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Representation

Biased codes are a way to represent values with a balanced number of positive and negative numbers using a pre-specified number N as a biasing value. Biased codes (and Gray codes) are non-weighted codes. In excess-3 code, numbers are represented as decimal digits, and each digit is represented by four bits as the digit value plus 3 (the "excess" amount):

• The smallest binary number represents the smallest value (0 − excess).
• The greatest binary number represents the largest value (2N+1 − excess − 1).

Excess-3, and Stibitz code

Decimal	Excess-3	Stibitz	BCD 8-4-2-1	Binary	3-of-6 CCITT extension1617	4-of-8 Hamming extension18
0	0011	0011	0000	0000	…10	…0011
1	0100	0100	0001	0001	…11	…1011
2	0101	0101	0010	0010	…10	…0101
3	0110	0110	0011	0011	…10	…0110
4	0111	0111	0100	0100	…00	…1000
5	1000	1000	0101	0101	…11	…0111
6	1001	1001	0110	0110	…10	…1001
7	1010	1010	0111	0111	…10	…1010
8	1011	1011	1000	1000	…00	…0100
9	1100	1100	1001	1001	…10	…1100

To encode a number such as 127, one simply encodes each of the decimal digits as above, giving (0100, 0101, 1010).

Excess-3 arithmetic uses different algorithms than normal non-biased BCD or binary positional system numbers. After adding two excess-3 digits, the raw sum is excess-6. For instance, after adding 1 (0100 in excess-3) and 2 (0101 in excess-3), the sum looks like 6 (1001 in excess-3) instead of 3 (0110 in excess-3). To correct this problem, after adding two digits, it is necessary to remove the extra bias by subtracting binary 0011 (decimal 3 in unbiased binary) if the resulting digit is less than decimal 10, or subtracting binary 1101 (decimal 13 in unbiased binary) if an overflow (carry) has occurred. (In 4-bit binary, subtracting binary 1101 is equivalent to adding 0011 and vice versa.)¹⁹

Advantage

The primary advantage of excess-3 coding over non-biased coding is that a decimal number can be nines' complemented²⁰ (for subtraction) as easily as a binary number can be ones' complemented: just by inverting all bits.²¹ Also, when the sum of two excess-3 digits is greater than 9, the carry bit of a 4-bit adder will be set high. This works because, after adding two digits, an "excess" value of 6 results in the sum. Because a 4-bit integer can only hold values 0 to 15, an excess of 6 means that any sum over 9 will overflow (produce a carry-out).

Another advantage is that the codes 0000 and 1111 are not used for any digit. A fault in a memory or basic transmission line may result in these codes. It is also more difficult to write the zero pattern to magnetic media.²²²³²⁴

Example

BCD 8-4-2-1 to excess-3 converter example in VHDL:

entity bcd8421xs3 is port ( a : in std_logic; b : in std_logic; c : in std_logic; d : in std_logic; an : buffer std_logic; bn : buffer std_logic; cn : buffer std_logic; dn : buffer std_logic; w : out std_logic; x : out std_logic; y : out std_logic; z : out std_logic ); end entity bcd8421xs3; architecture dataflow of bcd8421xs3 is begin an <= not a; bn <= not b; cn <= not c; dn <= not d; w <= (an and b and d ) or (a and bn and cn) or (an and b and c and dn); x <= (an and bn and d ) or (an and bn and c and dn) or (an and b and cn and dn) or (a and bn and cn and d); y <= (an and cn and dn) or (an and c and d ) or (a and bn and cn and dn); z <= (an and dn) or (a and bn and cn and dn); end architecture dataflow; -- of bcd8421xs3

Extensions

• 3-of-6 code extension: The excess-3 code is sometimes also used for data transfer, then often expanded to a 6-bit code per CCITT GT 43 No. 1, where 3 out of 6 bits are set.²⁵²⁶
• 4-of-8 code extension: As an alternative to the IBM transceiver code²⁷ (which is a 4-of-8 code with a Hamming distance of 2),²⁸ it is also possible to define a 4-of-8 excess-3 code extension achieving a Hamming distance of 4, if only denary digits are to be transferred.²⁹

See also

• Offset binary, excess-N, biased representation
• Excess-128
• Excess-Gray code
• Shifted Gray code
• Gray code
• m-of-n code
• Aiken code

References

1. Steinbuch, Karl W., ed. (1962). Written at Karlsruhe, Germany. Taschenbuch der Nachrichtenverarbeitung (in German) (1 ed.). Berlin / Göttingen / New York: Springer-Verlag OHG. pp. 71–73, 1081–1082. LCCN 62-14511. /wiki/Karl_W._Steinbuch ↩

2. Steinbuch, Karl W.; Weber, Wolfgang; Heinemann, Traute, eds. (1974) [1967]. Taschenbuch der Informatik – Band II – Struktur und Programmierung von EDV-Systemen (in German). Vol. 2 (3 ed.). Berlin, Germany: Springer Verlag. pp. 98–100. ISBN 3-540-06241-6. LCCN 73-80607. {{cite book}}: |work= ignored (help) 3-540-06241-6 ↩

3. Richards, Richard Kohler (1955). Arithmetic Operations in Digital Computers. New York, USA: van Nostrand. p. 182. /wiki/Van_Nostrand_(publisher) ↩

4. Kautz, William H. (June 1954). "Optimized Data Encoding for Digital Computers". Convention Record of the I.R.E. 1954 National Convention, Part 4: Electronic Computers and Information Technology. 2. Stanford Research Institute, Stanford, California, USA: The Institute of Radio Engineers, Inc.: 47–57. Session 19: Information Theory III - Speed and Computation. Retrieved 2020-05-22. (11 pages) /wiki/William_H._Kautz ↩

5. Steinbuch, Karl W., ed. (1962). Written at Karlsruhe, Germany. Taschenbuch der Nachrichtenverarbeitung (in German) (1 ed.). Berlin / Göttingen / New York: Springer-Verlag OHG. pp. 71–73, 1081–1082. LCCN 62-14511. /wiki/Karl_W._Steinbuch ↩

6. Schmid, Hermann (1974). Decimal Computation (1 ed.). Binghamton, New York, USA: John Wiley & Sons, Inc. p. 11. ISBN 0-471-76180-X. Retrieved 2016-01-03. 0-471-76180-X ↩

7. Schmid, Hermann (1983) [1974]. Decimal Computation (1 (reprint) ed.). Malabar, Florida, USA: Robert E. Krieger Publishing Company. p. 11. ISBN 0-89874-318-4. Retrieved 2016-01-03. (NB. At least some batches of this reprint edition were misprints with defective pages 115–146.) 0-89874-318-4 ↩

8. Stibitz, George Robert; Larrivee, Jules A. (1957). Written at Underhill, Vermont, USA. Mathematics and Computers (1 ed.). New York, USA / Toronto, Canada / London, UK: McGraw-Hill Book Company, Inc. p. 105. LCCN 56-10331. (10+228 pages) /wiki/George_Robert_Stibitz ↩

9. Steinbuch, Karl W., ed. (1962). Written at Karlsruhe, Germany. Taschenbuch der Nachrichtenverarbeitung (in German) (1 ed.). Berlin / Göttingen / New York: Springer-Verlag OHG. pp. 71–73, 1081–1082. LCCN 62-14511. /wiki/Karl_W._Steinbuch ↩

10. Steinbuch, Karl W.; Weber, Wolfgang; Heinemann, Traute, eds. (1974) [1967]. Taschenbuch der Informatik – Band II – Struktur und Programmierung von EDV-Systemen (in German). Vol. 2 (3 ed.). Berlin, Germany: Springer Verlag. pp. 98–100. ISBN 3-540-06241-6. LCCN 73-80607. {{cite book}}: |work= ignored (help) 3-540-06241-6 ↩

11. Dokter, Folkert; Steinhauer, Jürgen (1973-06-18). Digital Electronics. Philips Technical Library (PTL) / Macmillan Education (Reprint of 1st English ed.). Eindhoven, Netherlands: The Macmillan Press Ltd. / N. V. Philips' Gloeilampenfabrieken. pp. 42, 44. doi:10.1007/978-1-349-01417-0. ISBN 978-1-349-01419-4. SBN 333-13360-9. Retrieved 2018-07-01.[permanent dead link‍] (270 pages) (NB. This is based on a translation of volume I of the two-volume German edition.) 978-1-349-01419-4333-13360-9 ↩

12. Dokter, Folkert; Steinhauer, Jürgen (1975) [1969]. Digitale Elektronik in der Meßtechnik und Datenverarbeitung: Theoretische Grundlagen und Schaltungstechnik. Philips Fachbücher (in German). Vol. I (improved and extended 5th ed.). Hamburg, Germany: Deutsche Philips GmbH. pp. 48, 51, 53, 58, 61, 73. ISBN 3-87145-272-6. (xii+327+3 pages) (NB. The German edition of volume I was published in 1969, 1971, two editions in 1972, and 1975. Volume II was published in 1970, 1972, 1973, and 1975.) 3-87145-272-6 ↩

13. Stibitz, George Robert (1954-02-09) [1941-04-19]. "Complex Computer". Patent US2668661A. Retrieved 2020-05-24. [1] (102 pages) /wiki/George_Robert_Stibitz ↩

14. Mietke, Detlef (2017) [2015]. "Binäre Codices". Informations- und Kommunikationstechnik (in German). Berlin, Germany. Exzeß-3-Code mit Additions- und Subtraktionsverfahren. Archived from the original on 2017-04-25. Retrieved 2017-04-25. http://elektroniktutor.de/digitaltechnik/codices.html ↩

15. Ritchie, David (1986). The Computer Pioneers. New York, USA: Simon and Schuster. p. 35. ISBN 067152397X. 067152397X ↩

16. Comité Consultatif International Téléphonique et Télégraphique (CCITT), Groupe de Travail 43 (1959-06-03). Contribution No. 1. CCITT, GT 43 No. 1.{{cite book}}: CS1 maint: numeric names: authors list (link) /wiki/Comit%C3%A9_Consultatif_International_T%C3%A9l%C3%A9phonique_et_T%C3%A9l%C3%A9graphique ↩

17. Steinbuch, Karl W., ed. (1962). Written at Karlsruhe, Germany. Taschenbuch der Nachrichtenverarbeitung (in German) (1 ed.). Berlin / Göttingen / New York: Springer-Verlag OHG. pp. 71–73, 1081–1082. LCCN 62-14511. /wiki/Karl_W._Steinbuch ↩

18. Steinbuch, Karl W., ed. (1962). Written at Karlsruhe, Germany. Taschenbuch der Nachrichtenverarbeitung (in German) (1 ed.). Berlin / Göttingen / New York: Springer-Verlag OHG. pp. 71–73, 1081–1082. LCCN 62-14511. /wiki/Karl_W._Steinbuch ↩

19. Hayes, John P. (1978). Computer Architecture and Organization. McGraw-Hill International Book Company. p. 156. ISBN 0-07-027363-4. 0-07-027363-4 ↩

20. Steinbuch, Karl W., ed. (1962). Written at Karlsruhe, Germany. Taschenbuch der Nachrichtenverarbeitung (in German) (1 ed.). Berlin / Göttingen / New York: Springer-Verlag OHG. pp. 71–73, 1081–1082. LCCN 62-14511. /wiki/Karl_W._Steinbuch ↩

21. Steinbuch, Karl W., ed. (1962). Written at Karlsruhe, Germany. Taschenbuch der Nachrichtenverarbeitung (in German) (1 ed.). Berlin / Göttingen / New York: Springer-Verlag OHG. pp. 71–73, 1081–1082. LCCN 62-14511. /wiki/Karl_W._Steinbuch ↩

22. Steinbuch, Karl W., ed. (1962). Written at Karlsruhe, Germany. Taschenbuch der Nachrichtenverarbeitung (in German) (1 ed.). Berlin / Göttingen / New York: Springer-Verlag OHG. pp. 71–73, 1081–1082. LCCN 62-14511. /wiki/Karl_W._Steinbuch ↩

23. Bashe, Charles J.; Jackson, Peter Ward; Mussell, Howard A.; Winger, Wayne David (January 1956). "The Design of the IBM Type 702 System". Transactions of the American Institute of Electrical Engineers, Part I: Communication and Electronics. 74 (6): 695–704. doi:10.1109/TCE.1956.6372444. S2CID 51666209. Paper No. 55-719. /wiki/Transactions_of_the_American_Institute_of_Electrical_Engineers,_Part_I:_Communication_and_Electronics ↩

24. Mietke, Detlef (2017) [2015]. "Binäre Codices". Informations- und Kommunikationstechnik (in German). Berlin, Germany. Exzeß-3-Code mit Additions- und Subtraktionsverfahren. Archived from the original on 2017-04-25. Retrieved 2017-04-25. http://elektroniktutor.de/digitaltechnik/codices.html ↩

25. Comité Consultatif International Téléphonique et Télégraphique (CCITT), Groupe de Travail 43 (1959-06-03). Contribution No. 1. CCITT, GT 43 No. 1.{{cite book}}: CS1 maint: numeric names: authors list (link) /wiki/Comit%C3%A9_Consultatif_International_T%C3%A9l%C3%A9phonique_et_T%C3%A9l%C3%A9graphique ↩

26. Steinbuch, Karl W., ed. (1962). Written at Karlsruhe, Germany. Taschenbuch der Nachrichtenverarbeitung (in German) (1 ed.). Berlin / Göttingen / New York: Springer-Verlag OHG. pp. 71–73, 1081–1082. LCCN 62-14511. /wiki/Karl_W._Steinbuch ↩

27. IBM (July 1957). 65 Data Transceiver / 66 Printing Data Receiver. /wiki/IBM ↩

28. Steinbuch, Karl W., ed. (1962). Written at Karlsruhe, Germany. Taschenbuch der Nachrichtenverarbeitung (in German) (1 ed.). Berlin / Göttingen / New York: Springer-Verlag OHG. pp. 71–73, 1081–1082. LCCN 62-14511. /wiki/Karl_W._Steinbuch ↩

29. Steinbuch, Karl W., ed. (1962). Written at Karlsruhe, Germany. Taschenbuch der Nachrichtenverarbeitung (in German) (1 ed.). Berlin / Göttingen / New York: Springer-Verlag OHG. pp. 71–73, 1081–1082. LCCN 62-14511. /wiki/Karl_W._Steinbuch ↩