Two-out-of-five code

A two-out-of-five code is a <a href="/facts/Constant-weight_code/ftVAMFBD">constant-weight code</a> that provides exactly ten possible combinations of two bits, and is thus used for representing the <a href="/facts/Decimal_digit/JJdLBoGT">decimal digits</a> using five <a href="/facts/Bit/S972NSaD">bits</a>. Each bit is assigned a weight, such that the set bits sum to the desired value, with an exception for zero.
According to <a href="/facts/Federal_Standard_1037C/CA4jqghV">Federal Standard 1037C</a>:

<ul><li>each decimal <a href="/facts/Numerical_digit/JJdLBoGT">digit</a> is represented by a binary numeral consisting of five bits of which two are of one kind, called ones, and three are of the other kind, called zeros, and</li>
<li>the usual weights assigned to the bit positions are 0-1-2-3-6. However, in this scheme, <a href="/facts/0_(number)/4NBRtKHc">zero</a> is encoded as binary 01100; strictly speaking the 0-1-2-3-6 previously claimed is just a <a href="/facts/Mnemonic/6HfDn0ks">mnemonic</a> device.</li></ul>
The weights give a unique encoding for most digits, but allow two encodings for 3: 0+3 or 10010 and 1+2 or 01100. The former is used to encode the digit 3, and the latter is used to represent the otherwise unrepresentable zero.
The <a href="/facts/IBM_7070/fh2ammWI">IBM 7070</a>, <a href="/facts/IBM_7072/fh2ammWI">IBM 7072</a>, and <a href="/facts/IBM_7074/fh2ammWI">IBM 7074</a> computers used this code to represent each of the ten decimal digits in a machine word, although they numbered the bit positions 0-1-2-3-4, rather than with weights. Each word also had a sign flag, encoded using a two-out-of-three code, that could be A Alphanumeric, − Minus, or + Plus. When copied to a digit, the three bits were placed in bit positions 0-3-4. (Thus producing the numeric values 3, 6 and 9, respectively.)
A variant is the <a href="/facts/United_States_Postal_Service/1Je8qAHF">United States Postal Service</a> <a href="/facts/POSTNET/v4aGoqkP">POSTNET</a> <a href="/facts/Barcode/Mkh4cCTp">barcode</a>, used to represent the <a href="/facts/ZIP_Code/PHtdY0d4">ZIP Code</a> for automated mail sorting and routing equipment. This uses two tall bars as ones and three short bars as zeros. Here, the weights assigned to the bit positions are 7-4-2-1-0. Again, zero is encoded specially, using the 7+4 combination (binary 11000) that would naturally encode 11. This method was also used in North American telephone <a href="/facts/Multi-frequency/BvT5oGX3">multi-frequency</a> and <a href="/facts/Crossbar_switch/fSDXkMzf">crossbar switching</a> systems.
The USPS <a href="/facts/Postal_Alpha_Numeric_Encoding_Technique/4Gs6vuWI">Postal Alpha Numeric Encoding Technique</a> (PLANET) uses the same weights, but with the opposite bar-height convention.
The <a href="/facts/Code_39/49a4XzMm">Code 39</a> barcode uses weights 1-2-4-7-0 (i.e. <a href="/facts/Least_significant_bit/94h4P3qr">LSB</a> first, <a href="/facts/Parity_bit/peFp3Wau">Parity bit</a> last) for the widths of its bars, but it also encodes two bits of extra information in the spacing between bars. The || ||| spacing is used for digits.
The following table represents <a href="/facts/Decimal/7omfVNIM">decimal</a> digits from 0 to 9 in various two-out-of-five code systems:

<table><tbody><tr><th>Digit</th><th>Telecommunication 01236</th><th>POSTNET 74210</th><th>PLANET</th><th>Code39 bar widths 12470</th></tr><tr><th>1</th><td>11000</td><td>00011 </td><td>┃┃┃╻╻</td><td>▮ | | | ▮</td></tr><tr><th>2</th><td>10100</td><td>00101 </td><td>┃┃╻┃╻</td><td>| ▮ | | ▮</td></tr><tr><th>3</th><td>10010</td><td>00110 </td><td>┃┃╻╻┃</td><td>▮ ▮ | | |</td></tr><tr><th>4</th><td>01010</td><td>01001 </td><td>┃╻┃┃╻</td><td>| | ▮ | ▮</td></tr><tr><th>5</th><td>00110</td><td>01010 </td><td>┃╻┃╻┃</td><td>▮ | ▮ | |</td></tr><tr><th>6</th><td>10001</td><td>01100 </td><td>┃╻╻┃┃</td><td>| ▮ ▮ | |</td></tr><tr><th>7</th><td>01001</td><td>10001 </td><td>╻┃┃┃╻</td><td>| | | ▮ ▮</td></tr><tr><th>8</th><td>00101</td><td>10010 </td><td>╻┃┃╻┃</td><td>▮ | | ▮ |</td></tr><tr><th>9</th><td>00011</td><td>10100 </td><td>╻┃╻┃┃</td><td>| ▮ | ▮ |</td></tr><tr><th>0</th><td>01100</td><td>11000 </td><td>╻╻┃┃┃</td><td>| | ▮ ▮ |</td></tr><tr><td colspan="5"></td></tr><tr><th>A</th><td>1––10</td><th rowspan="3" colspan="3">IBM 707xSign flags</th></tr><tr><th>−</th><td>1––01</td></tr><tr><th>+</th><td>0––11</td></tr><tr><th>Digit</th><th>01234 IBM 707x</th><th colspan="3"></th></tr></tbody></table>
The requirement that exactly two bits be set is strictly stronger than a <a href="/facts/Parity_check/peFp3Wau">parity check</a>; like all <a href="/facts/Constant-weight_code/ftVAMFBD">constant-weight codes</a>, a two-out-of-five code can detect not only any single-bit error, but any <a href="/facts/Unidirectional_error/AcQHmD0x">unidirectional error</a> -- cases in which all the individual bit errors are of a single type (all 0→1 or all 1→0).

Two-out-of-five code open-in-new

Two-out-of-five code