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Excess-3, 3-excesscite-ref-steinbuch-1962-1-8[1]cite-ref-steinbuch-weber-1974-2-0[2]cite-ref-richards-1955-3-0[3] or 10-excess-3 binary code (often abbreviated as XS-3,cite-ref-kautz-1954-4-0[4] 3XScite-ref-steinbuch-1962-1-9[1] or X3cite-ref-schmid-1974-5-0[5]cite-ref-schmid-1983-6-0[6]), shifted binarycite-ref-stibitz-1957-7-0[7] or Stibitz codecite-ref-steinbuch-1962-1-10[1]cite-ref-steinbuch-weber-1974-2-1[2]cite-ref-dokter-1973-8-0[8]cite-ref-dokter-1975-9-0[9] (after George Stibitz,cite-ref-stibitz-1941-10-0[10] who built a relay-based adding machine in 1937cite-ref-mietke-2017-11-0[11]cite-ref-ritchie-1986-12-0[12]) is a self-complementary binary-coded decimal (BCD) code and numeral system. It is a biased representation. Excess-3 code was used on some older computers as well as in cash registers and hand-held portable electronic calculators of the 1970s, among other uses.

• Representation

• Advantage

• Example

• Extensions

• See also

• References

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).

| Decimal | Excess-3 | Stibitz | BCD 8-4-2-1 | Binary | 3-of-6 CCITT extension [ 13 ] [ 1 ] | 4-of-8 Hamming extension [ 1 ] |

|---|---|---|---|---|---|---|

| 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.)cite-ref-14[14]

The primary advantage of excess-3 coding over non-biased coding is that a decimal number can be nines' complementedcite-ref-steinbuch-1962-1-13[1] (for subtraction) as easily as a binary number can be ones' complemented: just by inverting all bits.cite-ref-steinbuch-1962-1-14[1] 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.cite-ref-steinbuch-1962-1-15[1]cite-ref-bashe-1956-15-0[15]cite-ref-mietke-2017-11-1[11]

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

• 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.cite-ref-ccitt-1959-13-1[13]cite-ref-steinbuch-1962-1-32[1]

• 4-of-8 code extension: As an alternative to the IBM transceiver codecite-ref-ibm-transceiver-16-0[16] (which is a 4-of-8 code with a Hamming distance of 2),cite-ref-steinbuch-1962-1-33[1] 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.cite-ref-steinbuch-1962-1-34[1]

• Offset binary, excess-N, biased representation

• Excess-128

• Excess-Gray code

• Shifted Gray code

• Gray code

• m-of-n code

• Aiken code

cite-note-steinbuch-1962-11. citerefsteinbuch1962Steinbuch, 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.

cite-note-steinbuch-weber-1974-22. ↑ citerefsteinbuchweberheinemann1974Steinbuch, 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)

cite-note-richards-1955-33. ↑ citerefrichards1955Richards, Richard Kohler (1955). Arithmetic Operations in Digital Computers. New York, USA: van Nostrand. p. 182.

cite-note-kautz-1954-44. ↑ citerefkautz1954Kautz, 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)

cite-note-schmid-1974-55. ↑ citerefschmid1974Schmid, 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.

cite-note-schmid-1983-66. ↑ citerefschmid1983Schmid, 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.)

cite-note-stibitz-1957-77. ↑ citerefstibitzlarrivee1957Stibitz, 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)

cite-note-dokter-1973-88. ↑ citerefdoktersteinhauer1973Dokter, 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. (270 pages) (NB. This is based on a translation of volume I of the two-volume German edition.)

cite-note-dokter-1975-99. ↑ citerefdoktersteinhauer1975Dokter, 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.)

cite-note-stibitz-1941-1010. ↑ citerefstibitz1954Stibitz, George Robert (1954-02-09) [1941-04-19]. "Complex Computer". Patent US2668661A. Retrieved 2020-05-24. [1] (102 pages)

cite-note-mietke-2017-1111. ↑ citerefmietke2017Mietke, 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.

cite-note-ritchie-1986-1212. ↑ citerefritchie1986Ritchie, David (1986). The Computer Pioneers. New York, USA: Simon and Schuster. p. 35. ISBN 067152397X.

cite-note-ccitt-1959-1313. citerefcomit-consultatif-international-t-l-phonique-et-t-l-graphique-ccitt-groupe-de-travail-431959Comité 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)

cite-note-1414. ↑ citerefhayes1978Hayes, John P. (1978). Computer Architecture and Organization. McGraw-Hill International Book Company. p. 156. ISBN 0-07-027363-4.

cite-note-bashe-1956-1515. ↑ citerefbashejacksonmussellwinger1956Bashe, 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.

cite-note-ibm-transceiver-1616. ↑ citerefibm1957IBM (July 1957). 65 Data Transceiver / 66 Printing Data Receiver.