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Chen–Ho encoding
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<h2 id="history">History</h2> <p>In what appears to have been a <a href="/facts/Multiple_discovery/8ISpCxDv">multiple discovery</a>, some of the concepts behind what later became known as Chen–Ho encoding were independently developed by Theodore M. Hertz in 1969<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> and by Tien Chi Chen (陳天機) (1928–)<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a><a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a><a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a><a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> in 1971. </p><p>Hertz of <a href="/facts/North_American_Rockwell_Corporation/dPSf6G1b">Rockwell</a> filed a patent for his encoding in 1969, which was granted in 1971.<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> </p><p>Chen first discussed his ideas with Irving Tze Ho (何宜慈) (1921–2003)<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a><a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a><a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a><a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a> in 1971. Chen and Ho were both working for <a href="/facts/IBM/QszA7nwd">IBM</a> at the time, albeit in different locations.<a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a><a class="footnote-ref" id="fnref:13" href="#fn:13"><sup>13</sup></a> Chen also consulted with Frank Chin Tung<a class="footnote-ref" id="fnref:14" href="#fn:14"><sup>14</sup></a> to verify the results of his theories independently.<a class="footnote-ref" id="fnref:15" href="#fn:15"><sup>15</sup></a> IBM filed a patent in their name in 1973, which was granted in 1974.<a class="footnote-ref" id="fnref:16" href="#fn:16"><sup>16</sup></a> At least by 1973, Hertz's earlier work must have been known to them, as the patent cites his patent as <a href="/facts/Prior_art/AEPs7QFA">prior art</a>.<a class="footnote-ref" id="fnref:17" href="#fn:17"><sup>17</sup></a> </p><p>With input from Joseph D. Rutledge and John C. McPherson,<a class="footnote-ref" id="fnref:18" href="#fn:18"><sup>18</sup></a> the final version of the Chen–Ho encoding was circulated inside IBM in 1974<a class="footnote-ref" id="fnref:19" href="#fn:19"><sup>19</sup></a> and published in 1975 in the journal <i><a href="/facts/Communications_of_the_ACM/Ix1DvEAk">Communications of the ACM</a></i>.<a class="footnote-ref" id="fnref:20" href="#fn:20"><sup>20</sup></a><a class="footnote-ref" id="fnref:21" href="#fn:21"><sup>21</sup></a> This version included several refinements, primarily related to the application of the encoding system. It constitutes a <a href="/facts/Huffmann_code/dBgMBz48">Huffman</a>-like <a href="/facts/Prefix_code/461neBYx">prefix code</a>. </p><p>The encoding was referred to as <i>Chen and Ho's scheme</i> in 1975,<a class="footnote-ref" id="fnref:22" href="#fn:22"><sup>22</sup></a> <i>Chen's encoding</i> in 1982<a class="footnote-ref" id="fnref:23" href="#fn:23"><sup>23</sup></a> and became known as <i>Chen–Ho encoding</i> or <i>Chen–Ho algorithm</i> since 2000.<a class="footnote-ref" id="fnref:24" href="#fn:24"><sup>24</sup></a> After having filed a patent for it in 2001,<a class="footnote-ref" id="fnref:25" href="#fn:25"><sup>25</sup></a> <a href="/facts/Michael_F._Cowlishaw/msVhIfWU">Michael F. Cowlishaw</a> published a further refinement of Chen–Ho encoding known as <a href="/facts/Densely_packed_decimal/DeSAEf3g">densely packed decimal</a> (DPD) encoding in <i>IEE Proceedings – Computers and Digital Techniques</i> in 2002.<a class="footnote-ref" id="fnref:26" href="#fn:26"><sup>26</sup></a><a class="footnote-ref" id="fnref:27" href="#fn:27"><sup>27</sup></a> DPD has subsequently been adopted as the <i>decimal encoding</i> used in the <a href="/facts/IEEE_754-2008/gqqJPoYC">IEEE 754-2008</a> and <a href="/facts/ISO/IEC/IEEE_60559:2011/dD3e7zAl">ISO/IEC/IEEE 60559:2011</a> <a href="/facts/Floating-point/eIckahxe">floating-point</a> standards. </p> <h2 id="application">Application</h2> <p>Chen noted that the digits zero through seven were simply encoded using three binary digits of the corresponding <a href="/facts/Octal/4nRcq0Ce">octal</a> group. He also postulated that one could use a <a href="/facts/Flag_(computing)/fWgTjqvX">flag</a> to identify a different encoding for the digits eight and nine, which would be encoded using a single bit. </p><p>In practice, a series of <a href="/facts/Boolean_algebra_(logic)/VkCToAmg">Boolean</a> transformations are applied to the stream of input bits, compressing BCD encoded digits from 12 bits per three digits to 10 bits per three digits. Reversed transformations are used to decode the resulting coded stream to BCD. Equivalent results can also be achieved by the use of a <a href="/facts/Look-up_table/ULPEAGTJ">look-up table</a>. </p><p>Chen–Ho encoding is limited to encoding sets of three decimal digits into groups of 10 bits (so called <i><a href="/facts/Declet_(computing)/lxjSrVJM">declets</a></i>).<a class="footnote-ref" id="fnref:28" href="#fn:28"><sup>28</sup></a> Of the 1024 states possible by using 10 bits, it leaves only 24 states unused<a class="footnote-ref" id="fnref:29" href="#fn:29"><sup>29</sup></a> (with <a href="/facts/Don%27t_care/kvQuwYMK">don't care</a> bits typically set to 0 on write and ignored on read). With only 2.34% wastage it gives a 20% more efficient encoding than BCD with one digit in 4 bits.<a class="footnote-ref" id="fnref:30" href="#fn:30"><sup>30</sup></a><a class="footnote-ref" id="fnref:31" href="#fn:31"><sup>31</sup></a> </p><p>Both, Hertz and Chen also proposed similar, but less efficient, encoding schemes to compress sets of two decimal digits (requiring 8 bits in BCD) into groups of 7 bits.<a class="footnote-ref" id="fnref:32" href="#fn:32"><sup>32</sup></a><a class="footnote-ref" id="fnref:33" href="#fn:33"><sup>33</sup></a> </p><p>Larger sets of decimal digits could be divided into three- and two-digit groups.<a class="footnote-ref" id="fnref:34" href="#fn:34"><sup>34</sup></a> </p><p>The patents also discuss the possibility to adapt the scheme to digits encoded in any other decimal codes than <a href="/facts/8-4-2-1_BCD/dWTaE7ES">8-4-2-1 BCD</a>,<a class="footnote-ref" id="fnref:35" href="#fn:35"><sup>35</sup></a> like f.e. <a href="/facts/Excess-3/shAvgBtQ">Excess-3</a>,<a class="footnote-ref" id="fnref:36" href="#fn:36"><sup>36</sup></a> <a href="/facts/Excess-6/4B81wxxe">Excess-6</a>, <a href="/facts/Jump-at-2_code/dWTaE7ES">Jump-at-2</a>, <a href="/facts/Jump-at-8_code/dWTaE7ES">Jump-at-8</a>, <a href="/facts/Gray_code/HtbVlM7T">Gray</a>, <a href="/facts/Glixon_code/HtbVlM7T">Glixon</a>, <a href="/facts/O%27Brien_I_code/HtbVlM7T">O'Brien type-I</a> and <a href="/facts/Gray%E2%80%93Stibitz_code/HtbVlM7T">Gray–Stibitz code</a>.<a class="footnote-ref" id="fnref:37" href="#fn:37"><sup>37</sup></a> The same principles could also be applied to other bases. </p><p>In 1973, some form of Chen–Ho encoding appears to have been utilized in the address conversion hardware of the optional <a href="/facts/IBM_7070/fh2ammWI">IBM 7070</a>/<a href="/facts/IBM_7074/fh2ammWI">7074</a> emulation feature for the <a href="/facts/IBM_System/370_Model_165/wnnjcRnT">IBM System/370 Model 165</a> and <a href="/facts/IBM_System/370_Model_168/xFls6Jyd">370 Model 168</a> computers.<a class="footnote-ref" id="fnref:38" href="#fn:38"><sup>38</sup></a><a class="footnote-ref" id="fnref:39" href="#fn:39"><sup>39</sup></a> </p><p>One prominent application uses a 128-bit register to store 33 decimal digits with a three digit exponent, effectively not less than what could be achieved using binary encoding (whereas BCD encoding would need 144 bits to store the same number of digits). </p> <h2 id="encodings-for-two-decimal-digits">Encodings for two decimal digits</h2> <h3>Hertz encoding</h3> Hertz decimal data encoding for a single <a href="/facts/Heptad_(computing)/lxjSrVJM">heptad</a> (1969 form)<a class="footnote-ref" id="fnref:40" href="#fn:40"><sup>40</sup></a><table><tbody><tr><th scope="col" colspan="8">Binary encoding</th><td rowspan="7"></td><th scope="col" colspan="5">Decimal digits</th></tr><tr><th scope="col">Code space (128 states)</th><th scope="col">b6</th><th scope="col">b5</th><th scope="col">b4</th><th scope="col">b3</th><th scope="col">b2</th><th scope="col">b1</th><th scope="col">b0</th><th>d1</th><th scope="col">d0</th><th scope="col">Values encoded</th><th scope="col">Description</th><th scope="col">Occurrences (100 states)</th></tr><tr><td>50% (64 states)</td><td>0</td><td>a</td><td>b</td><td>c</td><td>d</td><td>e</td><td>f</td><td>0abc</td><td>0def</td><td>(0–7) (0–7)</td><td>Two lower digits</td><td>64% (64 states)</td></tr><tr><td>12.5% (16 states)</td><td>1</td><td>1</td><td>0</td><td>c</td><td>d</td><td>e</td><td>f</td><td>100c</td><td>0def</td><td>(8–9) (0–7)</td><td rowspan="2">One lower digit,one higher digit</td><td>16% (16 states)</td></tr><tr><td>12.5% (16 states)</td><td>1</td><td>0</td><td>1</td><td>f</td><td>a</td><td>b</td><td>c</td><td>0abc</td><td>100f</td><td>(0–7) (8–9)</td><td>16% (16 states)</td></tr><tr><td>12.5% (16 states, 4 used)</td><td>1</td><td>1</td><td>1</td><td>c</td><td>x</td><td>x</td><td>f</td><td>100c</td><td>100f</td><td>(8–9) (8–9)</td><td>Two higher digits</td><td>4% (4 states)</td></tr><tr><td>12.5% (16 states, 0 used)</td><td>1</td><td>0</td><td>0</td><td>x</td><td>x</td><td>x</td><td>x</td><td></td><td></td><td></td><td></td><td>0% (0 states)</td></tr></tbody></table> <ul><li>This encoding is not <a href="/facts/Parity_(telecommunication)/peFp3Wau">parity</a>-preserving.</li></ul> <h3>Early Chen–Ho encoding, method A</h3> Decimal data encoding for a single <a href="/facts/Heptad_(computing)/lxjSrVJM">heptad</a> (early 1971 form, method A)<a class="footnote-ref" id="fnref:41" href="#fn:41"><sup>41</sup></a><table><tbody><tr><th scope="col" colspan="8">Binary encoding</th><td rowspan="6"></td><th scope="col" colspan="5">Decimal digits</th></tr><tr><th scope="col">Code space (128 states)</th><th scope="col">b6</th><th scope="col">b5</th><th scope="col">b4</th><th scope="col">b3</th><th scope="col">b2</th><th scope="col">b1</th><th scope="col">b0</th><th>d1</th><th scope="col">d0</th><th scope="col">Values encoded</th><th scope="col">Description</th><th scope="col">Occurrences (100 states)</th></tr><tr><td>50% (64 states)</td><td>0</td><td>a</td><td>b</td><td>c</td><td>d</td><td>e</td><td>f</td><td>0abc</td><td>0def</td><td>(0–7) (0–7)</td><td>Two lower digits</td><td>64% (64 states)</td></tr><tr><td>25% (32 states, 16 used)</td><td>1</td><td>0</td><td>x<a class="footnote-ref" id="fnref:42" href="#fn:42"><sup>42</sup></a> (b)<a class="footnote-ref" id="fnref:43" href="#fn:43"><sup>43</sup></a></td><td>c</td><td>d</td><td>e</td><td>f</td><td>100c</td><td>0def</td><td>(8–9) (0–7)</td><td rowspan="2">One lower digit,one higher digit</td><td>16% (16 states)</td></tr><tr><td>12.5% (16 states)</td><td>1</td><td>1</td><td>0</td><td>f</td><td>a</td><td>b</td><td>c</td><td>0abc</td><td>100f</td><td>(0–7) (8–9)</td><td>16% (16 states)</td></tr><tr><td>12.5% (16 states, 4 used)</td><td>1</td><td>1</td><td>1</td><td>c</td><td>x<a class="footnote-ref" id="fnref:44" href="#fn:44"><sup>44</sup></a> (a)<a class="footnote-ref" id="fnref:45" href="#fn:45"><sup>45</sup></a></td><td>x<a class="footnote-ref" id="fnref:46" href="#fn:46"><sup>46</sup></a> (b)<a class="footnote-ref" id="fnref:47" href="#fn:47"><sup>47</sup></a></td><td>f</td><td>100c</td><td>100f</td><td>(8–9) (8–9)</td><td>Two higher digits</td><td>4% (4 states)</td></tr></tbody></table> <ul><li>This encoding is not parity-preserving.</li></ul> <h3>Early Chen–Ho encoding, method B</h3> Decimal data encoding for a single heptad (early 1971 form, method B)<a class="footnote-ref" id="fnref:48" href="#fn:48"><sup>48</sup></a><table><tbody><tr><th scope="col" colspan="8">Binary encoding</th><td rowspan="7"></td><th scope="col" colspan="5">Decimal digits</th></tr><tr><th scope="col">Code space (128 states)</th><th scope="col">b6</th><th scope="col">b5</th><th scope="col">b4</th><th scope="col">b3</th><th scope="col">b2</th><th scope="col">b1</th><th scope="col">b0</th><th>d1</th><th scope="col">d0</th><th scope="col">Values encoded</th><th scope="col">Description</th><th scope="col">Occurrences (100 states)</th></tr><tr><td>50% (64 states)</td><td>0</td><td>a</td><td>b</td><td>c</td><td>d</td><td>e</td><td>f</td><td>0abc</td><td>0def</td><td>(0–7) (0–7)</td><td>Two lower digits</td><td>64% (64 states)</td></tr><tr><td>12.5% (16 states)</td><td>1</td><td>0</td><td>c</td><td>0</td><td>d</td><td>e</td><td>f</td><td>100c</td><td>0def</td><td>(8–9) (0–7)</td><td>One lower digit,one higher digit</td><td>16% (16 states)</td></tr><tr><td>12.5% (16 states, 4 used)</td><td>1</td><td>0</td><td>c</td><td>1</td><td>x</td><td>x</td><td>f</td><td>100c</td><td>100f</td><td>(8–9) (8–9)</td><td>Two higher digits</td><td>4% (4 states)</td></tr><tr><td>12.5% (16 states)</td><td>1</td><td>1</td><td>f</td><td>0</td><td>a</td><td>b</td><td>c</td><td>0abc</td><td>100f</td><td>(0–7) (8–9)</td><td>One lower digit,one higher digit</td><td>16% (16 states)</td></tr><tr><td>12.5% (16 states, 0 used)</td><td>1</td><td>1</td><td>x</td><td>1</td><td>x</td><td>x</td><td>x</td><td></td><td></td><td></td><td></td><td>0% (0 states)</td></tr></tbody></table> <ul><li>This encoding is not parity-preserving.</li></ul> <h3>Patented and final Chen–Ho encoding</h3> Decimal data encoding for a single heptad (patented 1973 form<a class="footnote-ref" id="fnref:49" href="#fn:49"><sup>49</sup></a> and final 1975 form<a class="footnote-ref" id="fnref:50" href="#fn:50"><sup>50</sup></a>)<table><tbody><tr><th scope="col" colspan="8">Binary encoding</th><td rowspan="6"></td><th scope="col" colspan="5">Decimal digits</th></tr><tr><th scope="col">Code space (128 states)</th><th scope="col">b6</th><th scope="col">b5</th><th scope="col">b4</th><th scope="col">b3</th><th scope="col">b2</th><th scope="col">b1</th><th scope="col">b0</th><th>d1</th><th scope="col">d0</th><th scope="col">Values encoded</th><th scope="col">Description</th><th scope="col">Occurrences (100 states)</th></tr><tr><td>50% (64 states)</td><td>0</td><td>a</td><td>b</td><td>c</td><td>d</td><td>e</td><td>f</td><td>0abc</td><td>0def</td><td>(0–7) (0–7)</td><td>Two lower digits</td><td>64% (64 states)</td></tr><tr><td>25.0% (32 states, 16 used)</td><td>1</td><td>0</td><td>x<a class="footnote-ref" id="fnref:51" href="#fn:51"><sup>51</sup></a> (b)<a class="footnote-ref" id="fnref:52" href="#fn:52"><sup>52</sup></a></td><td>c</td><td>d</td><td>e</td><td>f</td><td>100c</td><td>0def</td><td>(8–9) (0–7)</td><td rowspan="2">One lower digit,one higher digit</td><td>16% (16 states)</td></tr><tr><td>12.5% (16 states)</td><td>1</td><td>1</td><td>1</td><td>c</td><td>a</td><td>b</td><td>f</td><td>0abc</td><td>100f</td><td>(0–7) (8–9)</td><td>16% (16 states)</td></tr><tr><td>12.5% (16 states, 4 used)</td><td>1</td><td>1</td><td>0</td><td>c</td><td>x<a class="footnote-ref" id="fnref:53" href="#fn:53"><sup>53</sup></a> (a)<a class="footnote-ref" id="fnref:54" href="#fn:54"><sup>54</sup></a></td><td>x<a class="footnote-ref" id="fnref:55" href="#fn:55"><sup>55</sup></a> (b)<a class="footnote-ref" id="fnref:56" href="#fn:56"><sup>56</sup></a></td><td>f</td><td>100c</td><td>100f</td><td>(8–9) (8–9)</td><td>Two higher digits</td><td>4% (4 states)</td></tr></tbody></table> <ul><li>Assuming certain values for the <a href="/facts/Don%27t-care_(logic)/kvQuwYMK">don't-care</a> bits (f.e. 0), this encoding is <a href="/facts/Parity_(telecommunication)/peFp3Wau">parity</a>-preserving.<a class="footnote-ref" id="fnref:57" href="#fn:57"><sup>57</sup></a><a class="footnote-ref" id="fnref:58" href="#fn:58"><sup>58</sup></a></li></ul> <h2 id="encodings-for-three-decimal-digits">Encodings for three decimal digits</h2> <h3>Hertz encoding</h3> Hertz decimal data encoding for a single declet (1969 form)<a class="footnote-ref" id="fnref:59" href="#fn:59"><sup>59</sup></a><table><tbody><tr><th scope="col" colspan="11">Binary encoding</th><td rowspan="10"></td><th scope="col" colspan="6">Decimal digits</th></tr><tr><th scope="col">Code space (1024 states)</th><th scope="col">b9</th><th scope="col">b8</th><th scope="col">b7</th><th scope="col">b6</th><th scope="col">b5</th><th scope="col">b4</th><th scope="col">b3</th><th scope="col">b2</th><th scope="col">b1</th><th scope="col">b0</th><th scope="col">d2</th><th scope="col">d1</th><th scope="col">d0</th><th scope="col">Values encoded</th><th scope="col">Description</th><th scope="col">Occurrences (1000 states)</th></tr><tr><td>50.0% (512 states)</td><td>0</td><td>a</td><td>b</td><td>c</td><td>d</td><td>e</td><td>f</td><td>g</td><td>h</td><td>i</td><td>0abc</td><td>0def</td><td>0ghi</td><td>(0–7) (0–7) (0–7)</td><td>Three lower digits</td><td>51.2% (512 states)</td></tr><tr><td rowspan="3">37.5% (384 states)</td><td>1</td><td>0</td><td>0</td><td>c</td><td>d</td><td>e</td><td>f</td><td>g</td><td>h</td><td>i</td><td>100c</td><td>0def</td><td>0ghi</td><td>(8–9) (0–7) (0–7)</td><td rowspan="3">Two lower digits,one higher digit</td><td rowspan="3">38.4% (384 states)</td></tr><tr><td>1</td><td>0</td><td>1</td><td>f</td><td>a</td><td>b</td><td>c</td><td>g</td><td>h</td><td>i</td><td>0abc</td><td>100f</td><td>0ghi</td><td>(0–7) (8–9) (0–7)</td></tr><tr><td>1</td><td>1</td><td>0</td><td>i</td><td>a</td><td>b</td><td>c</td><td>d</td><td>e</td><td>f</td><td>0abc</td><td>0def</td><td>100i</td><td>(0–7) (0–7) (8–9)</td></tr><tr><td rowspan="3">9.375% (96 states)</td><td>1</td><td>1</td><td>1</td><td>f</td><td>0</td><td>0</td><td>i</td><td>a</td><td>b</td><td>c</td><td>0abc</td><td>100f</td><td>100i</td><td>(0–7) (8–9) (8–9)</td><td rowspan="3">One lower digit,two higher digits</td><td rowspan="3">9.6% (96 states)</td></tr><tr><td>1</td><td>1</td><td>1</td><td>c</td><td>0</td><td>1</td><td>i</td><td>d</td><td>e</td><td>f</td><td>100c</td><td>0def</td><td>100i</td><td>(8–9) (0–7) (8–9)</td></tr><tr><td>1</td><td>1</td><td>1</td><td>c</td><td>1</td><td>0</td><td>f</td><td>g</td><td>h</td><td>i</td><td>100c</td><td>100f</td><td>0ghi</td><td>(8–9) (8–9) (0–7)</td></tr><tr><td>3.125% (32 states, 8 used)</td><td>1</td><td>1</td><td>1</td><td>c</td><td>1</td><td>1</td><td>f</td><td>(0)</td><td>(0)</td><td>i</td><td>100c</td><td>100f</td><td>100i</td><td>(8–9) (8–9) (8–9)</td><td>Three higher digits, bits b2 and b1 are <a href="/facts/Don%27t_care/kvQuwYMK">don't care</a></td><td>0.8% (8 states)</td></tr></tbody></table> <ul><li>This encoding is not parity-preserving.</li></ul> <h3>Early Chen–Ho encoding</h3> Decimal data encoding for a single declet (early 1971 form)<a class="footnote-ref" id="fnref:60" href="#fn:60"><sup>60</sup></a><table><tbody><tr><th scope="col" colspan="11">Binary encoding</th><td rowspan="10"></td><th scope="col" colspan="6">Decimal digits</th></tr><tr><th scope="col">Code space (1024 states)</th><th scope="col">b9</th><th scope="col">b8</th><th scope="col">b7</th><th scope="col">b6</th><th scope="col">b5</th><th scope="col">b4</th><th scope="col">b3</th><th scope="col">b2</th><th scope="col">b1</th><th scope="col">b0</th><th scope="col">d2</th><th scope="col">d1</th><th scope="col">d0</th><th scope="col">Values encoded</th><th scope="col">Description</th><th scope="col">Occurrences (1000 states)</th></tr><tr><td>50.0% (512 states)</td><td>0</td><td>a</td><td>b</td><td>c</td><td>d</td><td>e</td><td>f</td><td>g</td><td>h</td><td>i</td><td>0abc</td><td>0def</td><td>0ghi</td><td>(0–7) (0–7) (0–7)</td><td>Three lower digits</td><td>51.2% (512 states)</td></tr><tr><td rowspan="3">37.5% (384 states)</td><td>1</td><td>0</td><td>0</td><td>c</td><td>d</td><td>e</td><td>f</td><td>g</td><td>h</td><td>i</td><td>100c</td><td>0def</td><td>0ghi</td><td>(8–9) (0–7) (0–7)</td><td rowspan="3">Two lower digits,one higher digit</td><td rowspan="3">38.4% (384 states)</td></tr><tr><td>1</td><td>0</td><td>1</td><td>f</td><td>g</td><td>h</td><td>i</td><td>a</td><td>b</td><td>c</td><td>0abc</td><td>100f</td><td>0ghi</td><td>(0–7) (8–9) (0–7)</td></tr><tr><td>1</td><td>1</td><td>0</td><td>i</td><td>a</td><td>b</td><td>c</td><td>d</td><td>e</td><td>f</td><td>0abc</td><td>0def</td><td>100i</td><td>(0–7) (0–7) (8–9)</td></tr><tr><td rowspan="3">9.375% (96 states)</td><td>1</td><td>1</td><td>1</td><td>0</td><td>0</td><td>f</td><td>i</td><td>a</td><td>b</td><td>c</td><td>0abc</td><td>100f</td><td>100i</td><td>(0–7) (8–9) (8–9)</td><td rowspan="3">One lower digit,two higher digits</td><td rowspan="3">9.6% (96 states)</td></tr><tr><td>1</td><td>1</td><td>1</td><td>0</td><td>1</td><td>i</td><td>c</td><td>d</td><td>e</td><td>f</td><td>100c</td><td>0def</td><td>100i</td><td>(8–9) (0–7) (8–9)</td></tr><tr><td>1</td><td>1</td><td>1</td><td>1</td><td>0</td><td>c</td><td>f</td><td>g</td><td>h</td><td>i</td><td>100c</td><td>100f</td><td>0ghi</td><td>(8–9) (8–9) (0–7)</td></tr><tr><td>3.125% (32 states, 8 used)</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>c</td><td>f</td><td>i</td><td>(0)</td><td>(0)</td><td>100c</td><td>100f</td><td>100i</td><td>(8–9) (8–9) (8–9)</td><td>Three higher digits, bits b1 and b0 are <a href="/facts/Don%27t_care/kvQuwYMK">don't care</a></td><td>0.8% (8 states)</td></tr></tbody></table> <ul><li>This encoding is not parity-preserving.</li></ul> <h3>Patented Chen–Ho encoding</h3> Decimal data encoding for a single declet (patented 1973 form)<a class="footnote-ref" id="fnref:61" href="#fn:61"><sup>61</sup></a><table><tbody><tr><th scope="col" colspan="11">Binary encoding</th><td rowspan="10"></td><th scope="col" colspan="6">Decimal digits</th></tr><tr><th scope="col">Code space (1024 states)</th><th scope="col">b9</th><th scope="col">b8</th><th scope="col">b7</th><th scope="col">b6</th><th scope="col">b5</th><th scope="col">b4</th><th scope="col">b3</th><th scope="col">b2</th><th scope="col">b1</th><th scope="col">b0</th><th scope="col">d2</th><th scope="col">d1</th><th scope="col">d0</th><th scope="col">Values encoded</th><th scope="col">Description</th><th scope="col">Occurrences (1000 states)</th></tr><tr><td>50.0% (512 states)</td><td>0</td><td>a</td><td>b</td><td>d</td><td>e</td><td>g</td><td>h</td><td>c</td><td>f</td><td>i</td><td>0abc</td><td>0def</td><td>0ghi</td><td>(0–7) (0–7) (0–7)</td><td>Three lower digits</td><td>51.2% (512 states)</td></tr><tr><td rowspan="3">37.5% (384 states)</td><td>1</td><td>0</td><td>0</td><td>d</td><td>e</td><td>g</td><td>h</td><td>c</td><td>f</td><td>i</td><td>100c</td><td>0def</td><td>0ghi</td><td>(8–9) (0–7) (0–7)</td><td rowspan="3">Two lower digits,one higher digit</td><td rowspan="3">38.4% (384 states)</td></tr><tr><td>1</td><td>0</td><td>1</td><td>a</td><td>b</td><td>g</td><td>h</td><td>c</td><td>f</td><td>i</td><td>0abc</td><td>100f</td><td>0ghi</td><td>(0–7) (8–9) (0–7)</td></tr><tr><td>1</td><td>1</td><td>0</td><td>d</td><td>e</td><td>a</td><td>b</td><td>c</td><td>f</td><td>i</td><td>0abc</td><td>0def</td><td>100i</td><td>(0–7) (0–7) (8–9)</td></tr><tr><td rowspan="3">9.375% (96 states)</td><td>1</td><td>1</td><td>1</td><td>1</td><td>0</td><td>a</td><td>b</td><td>c</td><td>f</td><td>i</td><td>0abc</td><td>100f</td><td>100i</td><td>(0–7) (8–9) (8–9)</td><td rowspan="3">One lower digit,two higher digits</td><td rowspan="3">9.6% (96 states)</td></tr><tr><td>1</td><td>1</td><td>1</td><td>0</td><td>1</td><td>d</td><td>e</td><td>c</td><td>f</td><td>i</td><td>100c</td><td>0def</td><td>100i</td><td>(8–9) (0–7) (8–9)</td></tr><tr><td>1</td><td>1</td><td>1</td><td>0</td><td>0</td><td>g</td><td>h</td><td>c</td><td>f</td><td>i</td><td>100c</td><td>100f</td><td>0ghi</td><td>(8–9) (8–9) (0–7)</td></tr><tr><td>3.125% (32 states, 8 used)</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>(0)</td><td>(0)</td><td>c</td><td>f</td><td>i</td><td>100c</td><td>100f</td><td>100i</td><td>(8–9) (8–9) (8–9)</td><td>Three higher digits, bits b4 and b3 are <a href="/facts/Don%27t_care/kvQuwYMK">don't care</a></td><td>0.8% (8 states)</td></tr></tbody></table> <ul><li>This encoding is not parity-preserving.<a class="footnote-ref" id="fnref:62" href="#fn:62"><sup>62</sup></a></li></ul> <h3>Final Chen–Ho encoding</h3> Chen-Ho decimal data encoding for a single declet (final 1975 form)<a class="footnote-ref" id="fnref:63" href="#fn:63"><sup>63</sup></a><a class="footnote-ref" id="fnref:64" href="#fn:64"><sup>64</sup></a><table><tbody><tr><th scope="col" colspan="11">Binary encoding</th><td rowspan="10"></td><th scope="col" colspan="6">Decimal digits</th></tr><tr><th scope="col">Code space (1024 states)</th><th scope="col">b9</th><th scope="col">b8</th><th scope="col">b7</th><th scope="col">b6</th><th scope="col">b5</th><th scope="col">b4</th><th scope="col">b3</th><th scope="col">b2</th><th scope="col">b1</th><th scope="col">b0</th><th scope="col">d2</th><th scope="col">d1</th><th scope="col">d0</th><th scope="col">Values encoded</th><th scope="col">Description</th><th scope="col">Occurrences (1000 states)</th></tr><tr><td>50.0% (512 states)</td><td>0</td><td>a</td><td>b</td><td>c</td><td>d</td><td>e</td><td>f</td><td>g</td><td>h</td><td>i</td><td>0abc</td><td>0def</td><td>0ghi</td><td>(0–7) (0–7) (0–7)</td><td>Three lower digits</td><td>51.2% (512 states)</td></tr><tr><td rowspan="3">37.5% (384 states)</td><td>1</td><td>0</td><td>0</td><td>c</td><td>d</td><td>e</td><td>f</td><td>g</td><td>h</td><td>i</td><td>100c</td><td>0def</td><td>0ghi</td><td>(8–9) (0–7) (0–7)</td><td rowspan="3">Two lower digits,one higher digit</td><td rowspan="3">38.4% (384 states)</td></tr><tr><td>1</td><td>0</td><td>1</td><td>c</td><td>a</td><td>b</td><td>f</td><td>g</td><td>h</td><td>i</td><td>0abc</td><td>100f</td><td>0ghi</td><td>(0–7) (8–9) (0–7)</td></tr><tr><td>1</td><td>1</td><td>0</td><td>c</td><td>d</td><td>e</td><td>f</td><td>a</td><td>b</td><td>i</td><td>0abc</td><td>0def</td><td>100i</td><td>(0–7) (0–7) (8–9)</td></tr><tr><td rowspan="3">9.375% (96 states)</td><td>1</td><td>1</td><td>1</td><td>c</td><td>0</td><td>0</td><td>f</td><td>a</td><td>b</td><td>i</td><td>0abc</td><td>100f</td><td>100i</td><td>(0–7) (8–9) (8–9)</td><td rowspan="3">One lower digit,two higher digits</td><td rowspan="3">9.6% (96 states)</td></tr><tr><td>1</td><td>1</td><td>1</td><td>c</td><td>0</td><td>1</td><td>f</td><td>d</td><td>e</td><td>i</td><td>100c</td><td>0def</td><td>100i</td><td>(8–9) (0–7) (8–9)</td></tr><tr><td>1</td><td>1</td><td>1</td><td>c</td><td>1</td><td>0</td><td>f</td><td>g</td><td>h</td><td>i</td><td>100c</td><td>100f</td><td>0ghi</td><td>(8–9) (8–9) (0–7)</td></tr><tr><td>3.125% (32 states, 8 used)</td><td>1</td><td>1</td><td>1</td><td>c</td><td>1</td><td>1</td><td>f</td><td>(0)</td><td>(0)</td><td>i</td><td>100c</td><td>100f</td><td>100i</td><td>(8–9) (8–9) (8–9)</td><td>Three higher digits, bits b2 and b1 are <a href="/facts/Don%27t_care/kvQuwYMK">don't care</a></td><td>0.8% (8 states)</td></tr></tbody></table> <ul><li>This encoding is not parity-preserving.<a class="footnote-ref" id="fnref:65" href="#fn:65"><sup>65</sup></a></li></ul> <h2 id="storage-efficiency">Storage efficiency</h2> Storage efficiency<table><tbody><tr><th colspan="4">BCD</th><th colspan="4">Necessary bits</th><th colspan="2">Bit difference</th></tr><tr><th>Digits</th><th>States</th><th>Bits</th><th>Binary code space</th><th>Binary encoding [A]</th><th>2-digit encoding [B]</th><th>3-digit encoding [C]</th><th>Mixed encoding</th><th>Mixed vs. Binary</th><th>Mixed vs. BCD</th></tr><tr><td>1</td><td>10</td><td>4</td><td>16</td><td>4</td><td>(7)</td><td>(10)</td><td>4 [1×A]</td><td>0</td><td>0</td></tr><tr><td>2</td><td>100</td><td>8</td><td>128</td><td>7</td><td>7</td><td>(10)</td><td>7 [1×B]</td><td>0</td><td>−1</td></tr><tr><td>3</td><td>1000</td><td>12</td><td>1024</td><td>10</td><td>(14)</td><td>10</td><td>10 [1×C]</td><td>0</td><td>−2</td></tr><tr><td>4</td><td>10000</td><td>16</td><td>16384</td><td>14</td><td>14</td><td>(20)</td><td>14 [2×B]</td><td>0</td><td>−2</td></tr><tr><td>5</td><td>100000</td><td>20</td><td>131072</td><td>17</td><td>(21)</td><td>(20)</td><td>17 [1×C+1×B]</td><td>0</td><td>−3</td></tr><tr><td>6</td><td>1000000</td><td>24</td><td>1048576</td><td>20</td><td>21</td><td>20</td><td>20 [2×C]</td><td>0</td><td>−4</td></tr><tr><td>7</td><td>10000000</td><td>28</td><td>16777216</td><td>24</td><td>(28)</td><td>(30)</td><td>24 [2×C+1×A]</td><td>0</td><td>−4</td></tr><tr><td>8</td><td>100000000</td><td>32</td><td>134217728</td><td>27</td><td>28</td><td>(30)</td><td>27 [2×C+1×B]</td><td>0</td><td>−5</td></tr><tr><td>9</td><td>1000000000</td><td>36</td><td>1073741824</td><td>30</td><td>(35)</td><td>30</td><td>30 [3×C]</td><td>0</td><td>−6</td></tr><tr><td>10</td><td>10000000000</td><td>40</td><td>17179869184</td><td>34</td><td>35</td><td>(40)</td><td>34 [3×C+1×A]</td><td>0</td><td>−6</td></tr><tr><td>11</td><td>100000000000</td><td>44</td><td>137438953472</td><td>37</td><td>(42)</td><td>(40)</td><td>37 [3×C+1×B]</td><td>0</td><td>−7</td></tr><tr><td>12</td><td>1000000000000</td><td>48</td><td>1099511627776</td><td>40</td><td>42</td><td>40</td><td>40 [4×C]</td><td>0</td><td>−8</td></tr><tr><td>13</td><td>10000000000000</td><td>52</td><td>17592186044416</td><td>44</td><td>(49)</td><td>(50)</td><td>44 [4×C+1×A]</td><td>0</td><td>−8</td></tr><tr><td>14</td><td>100000000000000</td><td>56</td><td>140737488355328</td><td>47</td><td>49</td><td>(50)</td><td>47 [4×C+1×B]</td><td>0</td><td>−9</td></tr><tr><td>15</td><td>1000000000000000</td><td>60</td><td>1125899906842624</td><td>50</td><td>(56)</td><td>50</td><td>50 [5×C]</td><td>0</td><td>−10</td></tr><tr><td>16</td><td>10000000000000000</td><td>64</td><td>18014398509481984</td><td>54</td><td>56</td><td>(60)</td><td>54 [5×C+1×A]</td><td>0</td><td>−10</td></tr><tr><td>17</td><td>100000000000000000</td><td>68</td><td>144115188075855872</td><td>57</td><td>(63)</td><td>(60)</td><td>57 [5×C+1×B]</td><td>0</td><td>−11</td></tr><tr><td>18</td><td>1000000000000000000</td><td>72</td><td>1152921504606846976</td><td>60</td><td>63</td><td>60</td><td>60 [6×C]</td><td>0</td><td>−12</td></tr><tr><td>19</td><td>10000000000000000000</td><td>76</td><td>18446744073709551616</td><td>64</td><td>(70)</td><td>(70)</td><td>64 [6×C+1×A]</td><td>0</td><td>−12</td></tr><tr><td>20</td><td>…</td><td>80</td><td>…</td><td>67</td><td>70</td><td>(70)</td><td>67 [6×C+1×B]</td><td>0</td><td>−13</td></tr><tr><td>21</td><td>…</td><td>84</td><td>…</td><td>70</td><td>(77)</td><td>70</td><td>70 [7×C]</td><td>0</td><td>−14</td></tr><tr><td>22</td><td>…</td><td>88</td><td>…</td><td>74</td><td>77</td><td>(80)</td><td>74 [7×C+1×A]</td><td>0</td><td>−14</td></tr><tr><td>23</td><td>…</td><td>92</td><td>…</td><td>77</td><td>(84)</td><td>(80)</td><td>77 [7×C+1×B]</td><td>0</td><td>−15</td></tr><tr><td>24</td><td>…</td><td>96</td><td>…</td><td>80</td><td>84</td><td>80</td><td>80 [8×C]</td><td>0</td><td>−16</td></tr><tr><td>25</td><td>…</td><td>100</td><td>…</td><td>84</td><td>(91)</td><td>(90)</td><td>84 [8×C+1×A]</td><td>0</td><td>−16</td></tr><tr><td>26</td><td>…</td><td>104</td><td>…</td><td>87</td><td>91</td><td>(90)</td><td>87 [8×C+1×B]</td><td>0</td><td>−17</td></tr><tr><td>27</td><td>…</td><td>108</td><td>…</td><td>90</td><td>(98)</td><td>90</td><td>90 [9×C]</td><td>0</td><td>−18</td></tr><tr><td>28</td><td>…</td><td>112</td><td>…</td><td>94</td><td>98</td><td>(100)</td><td>94 [9×C+1×A]</td><td>0</td><td>−18</td></tr><tr><td>29</td><td>…</td><td>116</td><td>…</td><td>97</td><td>(105)</td><td>(100)</td><td>97 [9×C+1×B]</td><td>0</td><td>−19</td></tr><tr><td>30</td><td>…</td><td>120</td><td>…</td><td>100</td><td>105</td><td>100</td><td>100 [10×C]</td><td>0</td><td>−20</td></tr><tr><td>31</td><td>…</td><td>124</td><td>…</td><td>103</td><td>(112)</td><td>(110)</td><td>104 [10×C+1×A]</td><td>+1</td><td>−20</td></tr><tr><td>32</td><td>…</td><td>128</td><td>…</td><td>107</td><td>112</td><td>(110)</td><td>107 [10×C+1×B]</td><td>0</td><td>−21</td></tr><tr><td>33</td><td>…</td><td>132</td><td>…</td><td>110</td><td>(119)</td><td>110</td><td>110 [11×C]</td><td>0</td><td>−22</td></tr><tr><td>34</td><td>…</td><td>136</td><td>…</td><td>113</td><td>119</td><td>(120)</td><td>114 [11×C+1×A]</td><td>+1</td><td>−22</td></tr><tr><td>35</td><td>…</td><td>140</td><td>…</td><td>117</td><td>(126)</td><td>(120)</td><td>117 [11×C+1×B]</td><td>0</td><td>−23</td></tr><tr><td>36</td><td>…</td><td>144</td><td>…</td><td>120</td><td>126</td><td>120</td><td>120 [12×C]</td><td>0</td><td>−24</td></tr><tr><td>37</td><td>…</td><td>148</td><td>…</td><td>123</td><td>(133)</td><td>(130)</td><td>124 [12×C+1×A]</td><td>+1</td><td>−24</td></tr><tr><td>38</td><td>…</td><td>152</td><td>…</td><td>127</td><td>133</td><td>(130)</td><td>127 [12×C+1×B]</td><td>0</td><td>−25</td></tr><tr><td>…</td><td>…</td><td>…</td><td>…</td><td>…</td><td>…</td><td>…</td><td>…</td><td>…</td><td>…</td></tr></tbody></table> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Binary-coded_decimal/dWTaE7ES">Binary-coded decimal</a> (BCD)</li> <li><a href="/facts/Densely_packed_decimal/DeSAEf3g">Densely packed decimal</a> (DPD)</li> <li><a href="/facts/DEC_RADIX_50/YyAcueyS">DEC RADIX 50</a> / <a href="/facts/DEC_MOD40/YyAcueyS">MOD40</a></li> <li><a href="/facts/IBM_SQUOZE/vpPlJNxL">IBM SQUOZE</a></li> <li><a href="/facts/Packed_BCD/dWTaE7ES">Packed BCD</a></li> <li><a href="/facts/Unicode_transformation_format/DDu3MfPV">Unicode transformation format</a> (UTF) (similar encoding scheme)</li> <li><a href="/facts/Length-limited_Huffman_code/dBgMBz48">Length-limited Huffman code</a></li></ul> <h2 id="notes">Notes</h2> <h2 id="further-reading">Further reading</h2> <ul><li>Bonten, Jo H. M. (2009-10-06) [2006-10-05]. <a href="http://home.kpn.nl/jhm.bonten/computers/bitsandbytes/wordsizes/ibmpde.htm#dense4">"Packed Decimal Encoding IEEE-754-2008"</a>. Geldrop, Netherlands. <a href="https://web.archive.org/web/20180711143550/http://home.kpn.nl/jhm.bonten/computers/bitsandbytes/wordsizes/ibmpde.htm#dense4">Archived</a> from the original on 2018-07-11. Retrieved 2018-07-11.</li> <li>Savard, John J. G. (2018) [2001]. <a href="http://www.quadibloc.com/crypto/mi060301.htm">"Base-26 Armor"</a>. <i>quadibloc</i>. <a href="https://web.archive.org/web/20180721041840/http://www.quadibloc.com/crypto/mi060301.htm">Archived</a> from the original on 2018-07-21. Retrieved 2018-07-21.</li> <li>Rinaldi, Russell G.; Moore, Brian B. (1967-03-21) [1964-06-30]. Written at Poughkeepsie & New Paltz, New York, USA. <a href="https://patents.google.com/patent/US3310786A">"Data compression/expansion and compressed data processing"</a> (Patent). New York, USA: <a href="/facts/International_Business_Machines_Corporation/QszA7nwd">International Business Machines Corporation</a> (IBM). US Patent US3310786A. Retrieved 2018-07-18 (60 pages) <a href="https://web.archive.org/web/20180719213000/https://patentimages.storage.googleapis.com/98/65/5e/63a98aac4b5184/US3310786.pdf">[11]</a>, Rinaldi, Russell G.; Moore, Brian B. (1969-05-20) [1967-01-19, 1964-06-30]. Written at Poughkeepsie & New Paltz, New York, USA. <a href="https://patents.google.com/patent/US3445641A">"Serial digital adder employing a compressed data format"</a> (Patent). New York, USA: <a href="/facts/International_Business_Machines_Corporation/QszA7nwd">International Business Machines Corporation</a> (IBM). US Patent US3445641A. Retrieved 2018-07-18 (40 pages) <a href="https://web.archive.org/web/20180719214333/https://patentimages.storage.googleapis.com/f8/84/2b/227f22291c1846/US3445641.pdf">[12]</a> and Rinaldi, Russell G.; Moore, Brian B. (1969-03-11) [1967-01-19, 1964-06-30]. Written at Poughkeepsie & New Paltz, New York, USA. <a href="https://patents.google.com/patent/US3432811A">"Data compression/expansion and compressed data processing"</a> (Patent). New York, USA: <a href="/facts/International_Business_Machines_Corporation/QszA7nwd">International Business Machines Corporation</a> (IBM). US Patent US3432811A. Retrieved 2018-07-18. (11 pages) <a href="https://web.archive.org/web/20180719213355/https://patentimages.storage.googleapis.com/6c/e8/d7/c1d354d8c9b947/US3432811.pdf">[13]</a> (NB. Three expired patents cited in both, the Hertz and Chen–Ho patents.)</li> <li>Bender, Richard R.; Galage, Dominick J. (August 1961). "Packing Mode Control". <i><a href="/facts/IBM_Technical_Disclosure_Bulletin/420UvtbT">IBM Technical Disclosure Bulletin</a></i>. 4 (3): 61–63.</li> <li>Tilem, J. Y. (December 1962). "Data Packing and Unpacking Means". <i><a href="/facts/IBM_Technical_Disclosure_Bulletin/420UvtbT">IBM Technical Disclosure Bulletin</a></i>. 5 (7): 48–49.</li> <li>Lengyel, E. J.; McMahon, R. F. (March 1967). <a href="https://priorart.ip.com/IPCOM/000092875#text">"Direct Decimal to Binary Address Generator For Small Memories"</a>. <i><a href="/facts/IBM_Technical_Disclosure_Bulletin/420UvtbT">IBM Technical Disclosure Bulletin</a></i>. 9 (10): 1347. Retrieved 2020-06-03.</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Muller, Jean-Michel; Brisebarre, Nicolas; de Dinechin, Florent; Jeannerod, Claude-Pierre; Lefèvre, Vincent; Melquiond, Guillaume; Revol, Nathalie; Stehlé, Damien; Torres, Serge (2010). Handbook of Floating-Point Arithmetic (1 ed.). Birkhäuser. doi:10.1007/978-0-8176-4705-6. ISBN 978-0-8176-4704-9. 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IBM San Jose Research Laboratory, San Jose, California, USA: IBM. <a href="/w/index.php?title=Tien_Chi_Chen&action=edit&redlink=1" target="_blank">/w/index.php?title=Tien_Chi_Chen&action=edit&redlink=1</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></p></li> <li id="fn:13"><p>Chen, Tien Chi (1971-03-29). Decimal Number Compression (PDF) (Internal memo to Irving Tze Ho). IBM San Jose Research Laboratory, San Jose, California, USA: IBM. pp. 1–4. Archived (PDF) from the original on 2012-10-17. Retrieved 2016-02-07. (4 pages) <a href="/w/index.php?title=Tien_Chi_Chen&action=edit&redlink=1" target="_blank">/w/index.php?title=Tien_Chi_Chen&action=edit&redlink=1</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></p></li> <li id="fn:14"><p>IBM资深专家Frank Tung博士8月4日来我校演讲 [IBM senior expert Dr. Frank Tung came to our school on 4 August to give a speech] (in Chinese and English). Guangzhou, China: South China University of Technology (SCUT). 2004-08-04. 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Communications of the ACM. 18 (1). IBM San Jose Research Laboratory, San Jose, California, USA & IBM Systems Products Division, Poughkeepsie/East Fishkill, New York, USA: Association for Computing Machinery: 49–52. doi:10.1145/360569.360660. ISSN 0001-0782. S2CID 14301378. (4 pages) <a href="/w/index.php?title=Tien_Chi_Chen&action=edit&redlink=1" target="_blank">/w/index.php?title=Tien_Chi_Chen&action=edit&redlink=1</a> <a href="#fnref:18" class="footnote-back-ref">↩</a></p></li> <li id="fn:19"><p>Chen, Tien Chi; Ho, Irving Tze (1974-06-25). "Storage-Efficient Representation of Decimal Data". Research Report RJ 1420 (Technical report). IBM San Jose Research Laboratory, San Jose, California, USA: IBM. <a href="/w/index.php?title=Tien_Chi_Chen&action=edit&redlink=1" target="_blank">/w/index.php?title=Tien_Chi_Chen&action=edit&redlink=1</a> <a href="#fnref:19" class="footnote-back-ref">↩</a></p></li> <li id="fn:20"><p>Chen, Tien Chi; Ho, Irving Tze (January 1975) [April 1974]. "Storage-Efficient Representation of Decimal Data". Communications of the ACM. 18 (1). IBM San Jose Research Laboratory, San Jose, California, USA & IBM Systems Products Division, Poughkeepsie/East Fishkill, New York, USA: Association for Computing Machinery: 49–52. doi:10.1145/360569.360660. ISSN 0001-0782. S2CID 14301378. (4 pages) <a href="/w/index.php?title=Tien_Chi_Chen&action=edit&redlink=1" target="_blank">/w/index.php?title=Tien_Chi_Chen&action=edit&redlink=1</a> <a href="#fnref:20" class="footnote-back-ref">↩</a></p></li> <li id="fn:21"><p>Cowlishaw, Michael Frederic (2014) [June 2000]. "A Summary of Chen-Ho Decimal Data encoding". IBM. Archived from the original on 2015-09-24. Retrieved 2016-02-07. <a href="/wiki/Michael_Frederic_Cowlishaw" target="_blank">/wiki/Michael_Frederic_Cowlishaw</a> <a href="#fnref:21" class="footnote-back-ref">↩</a></p></li> <li id="fn:22"><p>Smith, Alan Jay (August 1975) [April 1975]. "Comments on a paper by T. C. Chen and I. T. Ho". 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Retrieved 2016-02-07.{{cite journal}}: CS1 maint: DOI inactive as of December 2024 (link) (3 pages) <a href="/wiki/Michael_Frederic_Cowlishaw" target="_blank">/wiki/Michael_Frederic_Cowlishaw</a> <a href="#fnref:26" class="footnote-back-ref">↩</a></p></li> <li id="fn:27"><p>Cowlishaw, Michael Frederic (2007-02-13) [2000-10-03]. "A Summary of Densely Packed Decimal encoding". IBM. Archived from the original on 2015-09-24. Retrieved 2016-02-07. <a href="/wiki/Michael_Frederic_Cowlishaw" target="_blank">/wiki/Michael_Frederic_Cowlishaw</a> <a href="#fnref:27" class="footnote-back-ref">↩</a></p></li> <li id="fn:28"><p>Muller, Jean-Michel; Brisebarre, Nicolas; de Dinechin, Florent; Jeannerod, Claude-Pierre; Lefèvre, Vincent; Melquiond, Guillaume; Revol, Nathalie; Stehlé, Damien; Torres, Serge (2010). Handbook of Floating-Point Arithmetic (1 ed.). Birkhäuser. doi:10.1007/978-0-8176-4705-6. ISBN 978-0-8176-4704-9. LCCN 2009939668. <a href="978-0-8176-4704-9" target="_blank">978-0-8176-4704-9</a> <a href="#fnref:28" class="footnote-back-ref">↩</a></p></li> <li id="fn:29"><p>Muller, Jean-Michel; Brisebarre, Nicolas; de Dinechin, Florent; Jeannerod, Claude-Pierre; Lefèvre, Vincent; Melquiond, Guillaume; Revol, Nathalie; Stehlé, Damien; Torres, Serge (2010). Handbook of Floating-Point Arithmetic (1 ed.). Birkhäuser. doi:10.1007/978-0-8176-4705-6. ISBN 978-0-8176-4704-9. LCCN 2009939668. <a href="978-0-8176-4704-9" target="_blank">978-0-8176-4704-9</a> <a href="#fnref:29" class="footnote-back-ref">↩</a></p></li> <li id="fn:30"><p>Chen, Tien Chi (1971-03-29). Decimal Number Compression (PDF) (Internal memo to Irving Tze Ho). IBM San Jose Research Laboratory, San Jose, California, USA: IBM. pp. 1–4. Archived (PDF) from the original on 2012-10-17. Retrieved 2016-02-07. (4 pages) <a href="/w/index.php?title=Tien_Chi_Chen&action=edit&redlink=1" target="_blank">/w/index.php?title=Tien_Chi_Chen&action=edit&redlink=1</a> <a href="#fnref:30" class="footnote-back-ref">↩</a></p></li> <li id="fn:31"><p>Cowlishaw, Michael Frederic (2014) [June 2000]. "A Summary of Chen-Ho Decimal Data encoding". IBM. Archived from the original on 2015-09-24. Retrieved 2016-02-07. <a href="/wiki/Michael_Frederic_Cowlishaw" target="_blank">/wiki/Michael_Frederic_Cowlishaw</a> <a href="#fnref:31" class="footnote-back-ref">↩</a></p></li> <li id="fn:32"><p>Hertz, Theodore M. (1971-11-02) [1969-12-15]. "System for the compact storage of decimal numbers" (Patent). Whittier, California, USA: North American Rockwell Corporation. US Patent US3618047A. Retrieved 2018-07-18. (8 pages) [1][2] (NB. This expired patent discusses a coding system very similar to Chen-Ho, also cited as prior art in the Chen–Ho patent.) <a href="https://patents.google.com/patent/US3618047A" target="_blank">https://patents.google.com/patent/US3618047A</a> <a href="#fnref:32" class="footnote-back-ref">↩</a></p></li> <li id="fn:33"><p>Chen, Tien Chi (1971-03-29). Decimal Number Compression (PDF) (Internal memo to Irving Tze Ho). IBM San Jose Research Laboratory, San Jose, California, USA: IBM. pp. 1–4. Archived (PDF) from the original on 2012-10-17. Retrieved 2016-02-07. (4 pages) <a href="/w/index.php?title=Tien_Chi_Chen&action=edit&redlink=1" target="_blank">/w/index.php?title=Tien_Chi_Chen&action=edit&redlink=1</a> <a href="#fnref:33" class="footnote-back-ref">↩</a></p></li> <li id="fn:34"><p>Hertz, Theodore M. (1971-11-02) [1969-12-15]. "System for the compact storage of decimal numbers" (Patent). Whittier, California, USA: North American Rockwell Corporation. US Patent US3618047A. Retrieved 2018-07-18. (8 pages) [1][2] (NB. This expired patent discusses a coding system very similar to Chen-Ho, also cited as prior art in the Chen–Ho patent.) <a href="https://patents.google.com/patent/US3618047A" target="_blank">https://patents.google.com/patent/US3618047A</a> <a href="#fnref:34" class="footnote-back-ref">↩</a></p></li> <li id="fn:35"><p>Hertz, Theodore M. (1971-11-02) [1969-12-15]. "System for the compact storage of decimal numbers" (Patent). Whittier, California, USA: North American Rockwell Corporation. US Patent US3618047A. Retrieved 2018-07-18. (8 pages) [1][2] (NB. This expired patent discusses a coding system very similar to Chen-Ho, also cited as prior art in the Chen–Ho patent.) <a href="https://patents.google.com/patent/US3618047A" target="_blank">https://patents.google.com/patent/US3618047A</a> <a href="#fnref:35" class="footnote-back-ref">↩</a></p></li> <li id="fn:36"><p>Hertz, Theodore M. (1971-11-02) [1969-12-15]. "System for the compact storage of decimal numbers" (Patent). Whittier, California, USA: North American Rockwell Corporation. US Patent US3618047A. Retrieved 2018-07-18. (8 pages) [1][2] (NB. This expired patent discusses a coding system very similar to Chen-Ho, also cited as prior art in the Chen–Ho patent.) <a href="https://patents.google.com/patent/US3618047A" target="_blank">https://patents.google.com/patent/US3618047A</a> <a href="#fnref:36" class="footnote-back-ref">↩</a></p></li> <li id="fn:37"><p>Some 4-bit decimal codes are particularly well suited as alternatives to the 8-4-2-1 BCD code: Jump-at-8 code uses the same values for the ordered states 0 to 7, whereas in the Gray BCD and Glixon codes the values for the states 0 to 7 are still from the same set, but ordered differently (which, however, is transparent for the Hertz, Chen–Ho or densely packed decimal (DPD) encodings, as they pass through the bits unaltered). In these four codes, the most-significant bit can be used as a flag denoting "large" values. For the two "large" values, all but one bits remain static (the two middle bits are always zero for 8-4-2-1 and one for Jump-at-8 code, whilst for Gray BCD code one bit is set and the other cleared, whereas for Glixon code the two lower bits are always zero and one bit inverted, thus the two "large" values being transparently swapped), requiring only minor adaptations in the encoding. Three other codes can be conveniently split into groups of eight and two states as well, containing values from two ranges of consecutive bit patterns. In the case of the and Excess-6 BCD and Jump-at-2 codes, the most-significant bit can still be used to distinguish between the two groups, however, compared to the Jump-at-8 code, the group of small values now contains only two states and the larger group contains the eight larger values. In the case of the O'Brien type-I and Gray–Stibitz code, the next-most significant bit can serve as a flag bit instead, with the remaining bits again forming two groups of consecutive values. Therefore, these differences remain transparent for the encoding. <a href="/wiki/8-4-2-1_BCD_code" target="_blank">/wiki/8-4-2-1_BCD_code</a> <a href="#fnref:37" class="footnote-back-ref">↩</a></p></li> <li id="fn:38"><p>Savard, John J. G. (2018) [2007]. "Chen-Ho Encoding and Densely Packed Decimal". quadibloc. Archived from the original on 2018-07-03. Retrieved 2018-07-16. <a href="http://www.quadibloc.com/comp/cp020301.htm" target="_blank">http://www.quadibloc.com/comp/cp020301.htm</a> <a href="#fnref:38" class="footnote-back-ref">↩</a></p></li> <li id="fn:39"><p>7070/7074 Compatibility Feature for IBM System/370 Models 165, 165 II, and 168 (PDF) (2 ed.). IBM. June 1973 [1970]. GA22-6958-1 (File No. 5/370-13). Archived (PDF) from the original on 2018-07-22. Retrieved 2018-07-21. (31+5 pages) <a href="http://www.bitsavers.org/pdf/ibm/370/compatibility_feature/GA22-6958-1_707x_Compatibility_Feature_for_IBM-370_165_168.pdf" target="_blank">http://www.bitsavers.org/pdf/ibm/370/compatibility_feature/GA22-6958-1_707x_Compatibility_Feature_for_IBM-370_165_168.pdf</a> <a href="#fnref:39" class="footnote-back-ref">↩</a></p></li> <li id="fn:40"><p>Hertz, Theodore M. (1971-11-02) [1969-12-15]. "System for the compact storage of decimal numbers" (Patent). Whittier, California, USA: North American Rockwell Corporation. US Patent US3618047A. Retrieved 2018-07-18. (8 pages) [1][2] (NB. This expired patent discusses a coding system very similar to Chen-Ho, also cited as prior art in the Chen–Ho patent.) <a href="https://patents.google.com/patent/US3618047A" target="_blank">https://patents.google.com/patent/US3618047A</a> <a href="#fnref:40" class="footnote-back-ref">↩</a></p></li> <li id="fn:41"><p>Chen, Tien Chi (1971-03-29). Decimal Number Compression (PDF) (Internal memo to Irving Tze Ho). IBM San Jose Research Laboratory, San Jose, California, USA: IBM. pp. 1–4. Archived (PDF) from the original on 2012-10-17. Retrieved 2016-02-07. (4 pages) <a href="/w/index.php?title=Tien_Chi_Chen&action=edit&redlink=1" target="_blank">/w/index.php?title=Tien_Chi_Chen&action=edit&redlink=1</a> <a href="#fnref:41" class="footnote-back-ref">↩</a></p></li> <li id="fn:42"><p>Chen, Tien Chi (1971-03-29). Decimal Number Compression (PDF) (Internal memo to Irving Tze Ho). IBM San Jose Research Laboratory, San Jose, California, USA: IBM. pp. 1–4. Archived (PDF) from the original on 2012-10-17. Retrieved 2016-02-07. (4 pages) <a href="/w/index.php?title=Tien_Chi_Chen&action=edit&redlink=1" target="_blank">/w/index.php?title=Tien_Chi_Chen&action=edit&redlink=1</a> <a href="#fnref:42" class="footnote-back-ref">↩</a></p></li> <li id="fn:43"><p>Chen, Tien Chi; Ho, Irving Tze (January 1975) [April 1974]. "Storage-Efficient Representation of Decimal Data". Communications of the ACM. 18 (1). IBM San Jose Research Laboratory, San Jose, California, USA & IBM Systems Products Division, Poughkeepsie/East Fishkill, New York, USA: Association for Computing Machinery: 49–52. doi:10.1145/360569.360660. ISSN 0001-0782. S2CID 14301378. (4 pages) <a href="/w/index.php?title=Tien_Chi_Chen&action=edit&redlink=1" target="_blank">/w/index.php?title=Tien_Chi_Chen&action=edit&redlink=1</a> <a href="#fnref:43" class="footnote-back-ref">↩</a></p></li> <li id="fn:44"><p>Chen, Tien Chi (1971-03-29). Decimal Number Compression (PDF) (Internal memo to Irving Tze Ho). IBM San Jose Research Laboratory, San Jose, California, USA: IBM. pp. 1–4. Archived (PDF) from the original on 2012-10-17. Retrieved 2016-02-07. (4 pages) <a href="/w/index.php?title=Tien_Chi_Chen&action=edit&redlink=1" target="_blank">/w/index.php?title=Tien_Chi_Chen&action=edit&redlink=1</a> <a href="#fnref:44" class="footnote-back-ref">↩</a></p></li> <li id="fn:45"><p>Chen, Tien Chi; Ho, Irving Tze (January 1975) [April 1974]. "Storage-Efficient Representation of Decimal Data". Communications of the ACM. 18 (1). IBM San Jose Research Laboratory, San Jose, California, USA & IBM Systems Products Division, Poughkeepsie/East Fishkill, New York, USA: Association for Computing Machinery: 49–52. doi:10.1145/360569.360660. ISSN 0001-0782. S2CID 14301378. (4 pages) <a href="/w/index.php?title=Tien_Chi_Chen&action=edit&redlink=1" target="_blank">/w/index.php?title=Tien_Chi_Chen&action=edit&redlink=1</a> <a href="#fnref:45" class="footnote-back-ref">↩</a></p></li> <li id="fn:46"><p>Chen, Tien Chi (1971-03-29). Decimal Number Compression (PDF) (Internal memo to Irving Tze Ho). IBM San Jose Research Laboratory, San Jose, California, USA: IBM. pp. 1–4. Archived (PDF) from the original on 2012-10-17. Retrieved 2016-02-07. (4 pages) <a href="/w/index.php?title=Tien_Chi_Chen&action=edit&redlink=1" target="_blank">/w/index.php?title=Tien_Chi_Chen&action=edit&redlink=1</a> <a href="#fnref:46" class="footnote-back-ref">↩</a></p></li> <li id="fn:47"><p>Chen, Tien Chi; Ho, Irving Tze (January 1975) [April 1974]. "Storage-Efficient Representation of Decimal Data". Communications of the ACM. 18 (1). IBM San Jose Research Laboratory, San Jose, California, USA & IBM Systems Products Division, Poughkeepsie/East Fishkill, New York, USA: Association for Computing Machinery: 49–52. doi:10.1145/360569.360660. ISSN 0001-0782. S2CID 14301378. (4 pages) <a href="/w/index.php?title=Tien_Chi_Chen&action=edit&redlink=1" target="_blank">/w/index.php?title=Tien_Chi_Chen&action=edit&redlink=1</a> <a href="#fnref:47" class="footnote-back-ref">↩</a></p></li> <li id="fn:48"><p>Chen, Tien Chi (1971-03-29). Decimal Number Compression (PDF) (Internal memo to Irving Tze Ho). IBM San Jose Research Laboratory, San Jose, California, USA: IBM. pp. 1–4. Archived (PDF) from the original on 2012-10-17. Retrieved 2016-02-07. (4 pages) <a href="/w/index.php?title=Tien_Chi_Chen&action=edit&redlink=1" target="_blank">/w/index.php?title=Tien_Chi_Chen&action=edit&redlink=1</a> <a href="#fnref:48" class="footnote-back-ref">↩</a></p></li> <li id="fn:49"><p>Chen, Tien Chi; Ho, Irving Tze (1974-10-15) [1973-06-18]. Written at San Jose, California, USA & Poughkeepsie, New York, USA. "Binary coded decimal conversion apparatus" (Patent). Armonk, New York, USA: International Business Machines Corporation (IBM). US Patent US3842414A. Retrieved 2018-07-18. (14 pages) [5][6] (NB. This expired patent is about the Chen–Ho algorithm.) <a href="/w/index.php?title=Tien_Chi_Chen&action=edit&redlink=1" target="_blank">/w/index.php?title=Tien_Chi_Chen&action=edit&redlink=1</a> <a href="#fnref:49" class="footnote-back-ref">↩</a></p></li> <li id="fn:50"><p>Chen, Tien Chi; Ho, Irving Tze (January 1975) [April 1974]. "Storage-Efficient Representation of Decimal Data". Communications of the ACM. 18 (1). IBM San Jose Research Laboratory, San Jose, California, USA & IBM Systems Products Division, Poughkeepsie/East Fishkill, New York, USA: Association for Computing Machinery: 49–52. doi:10.1145/360569.360660. ISSN 0001-0782. S2CID 14301378. (4 pages) <a href="/w/index.php?title=Tien_Chi_Chen&action=edit&redlink=1" target="_blank">/w/index.php?title=Tien_Chi_Chen&action=edit&redlink=1</a> <a href="#fnref:50" class="footnote-back-ref">↩</a></p></li> <li id="fn:51"><p>Chen, Tien Chi; Ho, Irving Tze (1974-10-15) [1973-06-18]. Written at San Jose, California, USA & Poughkeepsie, New York, USA. "Binary coded decimal conversion apparatus" (Patent). Armonk, New York, USA: International Business Machines Corporation (IBM). US Patent US3842414A. Retrieved 2018-07-18. (14 pages) [5][6] (NB. This expired patent is about the Chen–Ho algorithm.) <a href="/w/index.php?title=Tien_Chi_Chen&action=edit&redlink=1" target="_blank">/w/index.php?title=Tien_Chi_Chen&action=edit&redlink=1</a> <a href="#fnref:51" class="footnote-back-ref">↩</a></p></li> <li id="fn:52"><p>Chen, Tien Chi; Ho, Irving Tze (January 1975) [April 1974]. "Storage-Efficient Representation of Decimal Data". Communications of the ACM. 18 (1). IBM San Jose Research Laboratory, San Jose, California, USA & IBM Systems Products Division, Poughkeepsie/East Fishkill, New York, USA: Association for Computing Machinery: 49–52. doi:10.1145/360569.360660. ISSN 0001-0782. S2CID 14301378. (4 pages) <a href="/w/index.php?title=Tien_Chi_Chen&action=edit&redlink=1" target="_blank">/w/index.php?title=Tien_Chi_Chen&action=edit&redlink=1</a> <a href="#fnref:52" class="footnote-back-ref">↩</a></p></li> <li id="fn:53"><p>Chen, Tien Chi; Ho, Irving Tze (1974-10-15) [1973-06-18]. Written at San Jose, California, USA & Poughkeepsie, New York, USA. "Binary coded decimal conversion apparatus" (Patent). Armonk, New York, USA: International Business Machines Corporation (IBM). US Patent US3842414A. Retrieved 2018-07-18. (14 pages) [5][6] (NB. This expired patent is about the Chen–Ho algorithm.) <a href="/w/index.php?title=Tien_Chi_Chen&action=edit&redlink=1" target="_blank">/w/index.php?title=Tien_Chi_Chen&action=edit&redlink=1</a> <a href="#fnref:53" class="footnote-back-ref">↩</a></p></li> <li id="fn:54"><p>Chen, Tien Chi; Ho, Irving Tze (January 1975) [April 1974]. "Storage-Efficient Representation of Decimal Data". Communications of the ACM. 18 (1). IBM San Jose Research Laboratory, San Jose, California, USA & IBM Systems Products Division, Poughkeepsie/East Fishkill, New York, USA: Association for Computing Machinery: 49–52. doi:10.1145/360569.360660. ISSN 0001-0782. S2CID 14301378. (4 pages) <a href="/w/index.php?title=Tien_Chi_Chen&action=edit&redlink=1" target="_blank">/w/index.php?title=Tien_Chi_Chen&action=edit&redlink=1</a> <a href="#fnref:54" class="footnote-back-ref">↩</a></p></li> <li id="fn:55"><p>Chen, Tien Chi; Ho, Irving Tze (1974-10-15) [1973-06-18]. Written at San Jose, California, USA & Poughkeepsie, New York, USA. "Binary coded decimal conversion apparatus" (Patent). Armonk, New York, USA: International Business Machines Corporation (IBM). US Patent US3842414A. Retrieved 2018-07-18. (14 pages) [5][6] (NB. This expired patent is about the Chen–Ho algorithm.) <a href="/w/index.php?title=Tien_Chi_Chen&action=edit&redlink=1" target="_blank">/w/index.php?title=Tien_Chi_Chen&action=edit&redlink=1</a> <a href="#fnref:55" class="footnote-back-ref">↩</a></p></li> <li id="fn:56"><p>Chen, Tien Chi; Ho, Irving Tze (January 1975) [April 1974]. "Storage-Efficient Representation of Decimal Data". Communications of the ACM. 18 (1). IBM San Jose Research Laboratory, San Jose, California, USA & IBM Systems Products Division, Poughkeepsie/East Fishkill, New York, USA: Association for Computing Machinery: 49–52. doi:10.1145/360569.360660. ISSN 0001-0782. S2CID 14301378. (4 pages) <a href="/w/index.php?title=Tien_Chi_Chen&action=edit&redlink=1" target="_blank">/w/index.php?title=Tien_Chi_Chen&action=edit&redlink=1</a> <a href="#fnref:56" class="footnote-back-ref">↩</a></p></li> <li id="fn:57"><p>Chen, Tien Chi; Ho, Irving Tze (1974-10-15) [1973-06-18]. Written at San Jose, California, USA & Poughkeepsie, New York, USA. "Binary coded decimal conversion apparatus" (Patent). Armonk, New York, USA: International Business Machines Corporation (IBM). US Patent US3842414A. Retrieved 2018-07-18. (14 pages) [5][6] (NB. This expired patent is about the Chen–Ho algorithm.) <a href="/w/index.php?title=Tien_Chi_Chen&action=edit&redlink=1" target="_blank">/w/index.php?title=Tien_Chi_Chen&action=edit&redlink=1</a> <a href="#fnref:57" class="footnote-back-ref">↩</a></p></li> <li id="fn:58"><p>Chen, Tien Chi; Ho, Irving Tze (January 1975) [April 1974]. "Storage-Efficient Representation of Decimal Data". Communications of the ACM. 18 (1). IBM San Jose Research Laboratory, San Jose, California, USA & IBM Systems Products Division, Poughkeepsie/East Fishkill, New York, USA: Association for Computing Machinery: 49–52. doi:10.1145/360569.360660. ISSN 0001-0782. S2CID 14301378. (4 pages) <a href="/w/index.php?title=Tien_Chi_Chen&action=edit&redlink=1" target="_blank">/w/index.php?title=Tien_Chi_Chen&action=edit&redlink=1</a> <a href="#fnref:58" class="footnote-back-ref">↩</a></p></li> <li id="fn:59"><p>Hertz, Theodore M. (1971-11-02) [1969-12-15]. "System for the compact storage of decimal numbers" (Patent). Whittier, California, USA: North American Rockwell Corporation. US Patent US3618047A. Retrieved 2018-07-18. (8 pages) [1][2] (NB. This expired patent discusses a coding system very similar to Chen-Ho, also cited as prior art in the Chen–Ho patent.) <a href="https://patents.google.com/patent/US3618047A" target="_blank">https://patents.google.com/patent/US3618047A</a> <a href="#fnref:59" class="footnote-back-ref">↩</a></p></li> <li id="fn:60"><p>Chen, Tien Chi (1971-03-29). Decimal Number Compression (PDF) (Internal memo to Irving Tze Ho). IBM San Jose Research Laboratory, San Jose, California, USA: IBM. pp. 1–4. Archived (PDF) from the original on 2012-10-17. Retrieved 2016-02-07. (4 pages) <a href="/w/index.php?title=Tien_Chi_Chen&action=edit&redlink=1" target="_blank">/w/index.php?title=Tien_Chi_Chen&action=edit&redlink=1</a> <a href="#fnref:60" class="footnote-back-ref">↩</a></p></li> <li id="fn:61"><p>Chen, Tien Chi; Ho, Irving Tze (1974-10-15) [1973-06-18]. Written at San Jose, California, USA & Poughkeepsie, New York, USA. "Binary coded decimal conversion apparatus" (Patent). Armonk, New York, USA: International Business Machines Corporation (IBM). US Patent US3842414A. Retrieved 2018-07-18. (14 pages) [5][6] (NB. This expired patent is about the Chen–Ho algorithm.) <a href="/w/index.php?title=Tien_Chi_Chen&action=edit&redlink=1" target="_blank">/w/index.php?title=Tien_Chi_Chen&action=edit&redlink=1</a> <a href="#fnref:61" class="footnote-back-ref">↩</a></p></li> <li id="fn:62"><p>Chen, Tien Chi; Ho, Irving Tze (1974-10-15) [1973-06-18]. Written at San Jose, California, USA & Poughkeepsie, New York, USA. "Binary coded decimal conversion apparatus" (Patent). Armonk, New York, USA: International Business Machines Corporation (IBM). US Patent US3842414A. Retrieved 2018-07-18. (14 pages) [5][6] (NB. This expired patent is about the Chen–Ho algorithm.) <a href="/w/index.php?title=Tien_Chi_Chen&action=edit&redlink=1" target="_blank">/w/index.php?title=Tien_Chi_Chen&action=edit&redlink=1</a> <a href="#fnref:62" class="footnote-back-ref">↩</a></p></li> <li id="fn:63"><p>Chen, Tien Chi; Ho, Irving Tze (January 1975) [April 1974]. "Storage-Efficient Representation of Decimal Data". Communications of the ACM. 18 (1). IBM San Jose Research Laboratory, San Jose, California, USA & IBM Systems Products Division, Poughkeepsie/East Fishkill, New York, USA: Association for Computing Machinery: 49–52. doi:10.1145/360569.360660. ISSN 0001-0782. S2CID 14301378. (4 pages) <a href="/w/index.php?title=Tien_Chi_Chen&action=edit&redlink=1" target="_blank">/w/index.php?title=Tien_Chi_Chen&action=edit&redlink=1</a> <a href="#fnref:63" class="footnote-back-ref">↩</a></p></li> <li id="fn:64"><p>Cowlishaw, Michael Frederic (2014) [June 2000]. "A Summary of Chen-Ho Decimal Data encoding". IBM. Archived from the original on 2015-09-24. Retrieved 2016-02-07. <a href="/wiki/Michael_Frederic_Cowlishaw" target="_blank">/wiki/Michael_Frederic_Cowlishaw</a> <a href="#fnref:64" class="footnote-back-ref">↩</a></p></li> <li id="fn:65"><p>Chen, Tien Chi; Ho, Irving Tze (January 1975) [April 1974]. "Storage-Efficient Representation of Decimal Data". Communications of the ACM. 18 (1). IBM San Jose Research Laboratory, San Jose, California, USA & IBM Systems Products Division, Poughkeepsie/East Fishkill, New York, USA: Association for Computing Machinery: 49–52. doi:10.1145/360569.360660. ISSN 0001-0782. S2CID 14301378. (4 pages) <a href="/w/index.php?title=Tien_Chi_Chen&action=edit&redlink=1" target="_blank">/w/index.php?title=Tien_Chi_Chen&action=edit&redlink=1</a> <a href="#fnref:65" class="footnote-back-ref">↩</a></p></li> </ol>