Excess-3, 3-excess or 10-excess-3 binary code (often abbreviated as XS-3, 3XS or X3), shifted binary or Stibitz code (after George Stibitz, who built a relay-based adding machine in 1937) 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.
2 − excess − 1).
Excess-3, and Stibitz code
| Decimal |
Excess-3 |
Stibitz |
BCD 8-4-2-1 |
Binary |
3-of-6 CCITT
extension |
4-of-8 Hamming
extension |
| 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.)
The primary advantage of excess-3 coding over non-biased coding is that a decimal number can be nines' complemented (for subtraction) as easily as a binary number can be ones' complemented: just by inverting all bits. 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.
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.
- 4-of-8 code extension: As an alternative to the IBM transceiver code (which is a 4-of-8 code with a Hamming distance of 2), 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 book |title=Taschenbuch der Nachrichtenverarbeitung |language=de |editor-first=Karl W. |editor-last=Steinbuch |editor-link=Karl W. Steinbuch |date=1962 |edition=1 |publisher=[[Springer-Verlag OHG]] |location=Karlsruhe, Germany |publication-place=Berlin / Göttingen / New York |lccn=62-14511 |pages=71–73, 1081–1082}}</ref>\n<ref name=\"Steinbuch-Weber_1974\">{{cite book |title=Taschenbuch der Informatik – Band II – Struktur und Programmierung von EDV-Systemen |language=de |editor-first1=Karl W. |editor-last1=Steinbuch |editor-link1=Karl W. Steinbuch |editor-first2=Wolfgang |editor-last2=Weber |editor-first3=Traute |editor-last3=Heinemann |date=1974 |orig-year=1967 |edition=3 |volume=2 |work=Taschenbuch der Nachrichtenverarbeitung |publisher=[[Springer Verlag]] |location=Berlin, Germany |isbn=3-540-06241-6 |lccn=73-80607 |pages=98–100}}</ref>\n<ref name=\"Schmid_1974\">{{cite book |title=Decimal Computation |author-first=Hermann |author-last=Schmid<!-- General Electric Company, Binghamton, New York, USA --> |author-link=Hermann Schmid (computer scientist) |date=1974 |edition=1 |publisher=[[John Wiley & Sons, Inc.]] |location=Binghamton, New York, USA |isbn=0-471-76180-X |page=[https://archive.org/details/decimalcomputati0000schm/page/11 11] |url=https://archive.org/details/decimalcomputati0000schm |url-access=registration |access-date=2016-01-03}}</ref>\n<ref name=\"Schmid_1983\">{{cite book |title=Decimal Computation |author-first=Hermann |author-last=Schmid<!-- General Electric Company, Binghamton, New York, USA --> |author-link=Hermann Schmid (computer scientist) |orig-year=1974 |date=1983 |edition=1 (reprint) |publisher=Robert E. Krieger Publishing Company |location=Malabar, Florida, USA |isbn=0-89874-318-4 |page=11 |url=https://books.google.com/books?id=uEYZAQAAIAAJ |access-date=2016-01-03}} (NB. At least some batches of this reprint edition were [[misprint]]s with defective pages 115–146.<!-- they contain the contents of another book -->)</ref>\n<ref name=\"Richards_1955\">{{cite book |author-first=Richard Kohler |author-last=Richards |title=Arithmetic Operations in Digital Computers |publisher=[[van Nostrand (publisher)|van Nostrand]] |location=New York, USA |date=1955 |page=182}}</ref>\n<ref name=\"CCITT_1959\">{{cite book |author=[[Comité Consultatif International Téléphonique et Télégraphique]] (CCITT), Groupe de Travail 43 |title=Contribution No. 1 |date=1959-06-03 |id=CCITT, GT 43 No. 1}}</ref>\n<ref name=\"Bashe_1956\">{{cite journal |author-first1=Charles J. |author-last1=Bashe |author-first2=Peter Ward |author-last2=Jackson |author-first3=Howard A. |author-last3=Mussell |author-first4=Wayne David |author-last4=Winger |title=The Design of the IBM Type 702 System |journal=[[Transactions of the American Institute of Electrical Engineers, Part I: Communication and Electronics]] |volume=74 |issue=6 |date=January 1956 |pages=695–704 |doi=10.1109/TCE.1956.6372444 |s2cid=51666209 |id=Paper No. 55-719}}</ref>\n<ref name=\"IBM_Transceiver\">{{cite book |author=IBM |title=65 Data Transceiver / 66 Printing Data Receiver |date=July 1957 |author-link=IBM}}</ref>\n<ref name=\"Mietke_2017\">{{cite web |title=Binäre Codices |at=Exzeß-3-Code mit Additions- und Subtraktionsverfahren |work=Informations- und Kommunikationstechnik |language=de |author-first=Detlef |author-last=Mietke |location=Berlin, Germany |date=2017 |orig-year=2015 |url=http://elektroniktutor.de/digitaltechnik/codices.html |access-date=2017-04-25 |url-status=live |archive-url=https://web.archive.org/web/20170425181921/http://elektroniktutor.de/digitaltechnik/codices.html |archive-date=2017-04-25}}</ref>\n<ref name=\"Ritchie_1986\">{{cite book |author-last=Ritchie |author-first=David |date=1986 |title=The Computer Pioneers |page=[https://archive.org/details/computerpioneers00ritc/page/35 35] |location=New York, USA |publisher=[[Simon and Schuster]] |isbn=067152397X |url=https://archive.org/details/computerpioneers00ritc/page/35 }}</ref>\n<ref name=\"Dokter_1973\">{{cite book |title=Digital Electronics |author-first1=Folkert |author-last1=Dokter |author-first2=Jürgen |author-last2=Steinhauer |date=1973-06-18 |series=Philips Technical Library (PTL) / Macmillan Education |publisher=[[The Macmillan Press Ltd.]] / [[N. V. Philips' Gloeilampenfabrieken]] |edition=Reprint of 1st English |location=Eindhoven, Netherlands |sbn=333-13360-9 |isbn=978-1-349-01419-4 |doi=10.1007/978-1-349-01417-0 |pages=42, 44 |url=https://books.google.com/books?id=hlRdDwAAQBAJ |access-date=2018-07-01 }}{{Dead link|date=March 2024 |bot=InternetArchiveBot |fix-attempted=yes }} (270 pages) (NB. This is based on a translation of volume I of the two-volume German edition.)</ref>\n<ref name=\"Dokter_1975\">{{cite book |author-first1=Folkert |author-last1=Dokter |author-first2=Jürgen |author-last2=Steinhauer |title=Digitale Elektronik in der Meßtechnik und Datenverarbeitung: Theoretische Grundlagen und Schaltungstechnik |language=de |series=Philips Fachbücher |publisher=[[Deutsche Philips GmbH]] |location=Hamburg, Germany |volume=I |date=1975 |orig-year=1969 |edition=improved and extended 5th |isbn=3-87145-272-6 |pages=48, 51, 53, 58, 61, 73}} (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.)</ref>\n<ref name=\"Kautz_1954\">{{cite journal |author-first=William H. |author-last=Kautz |author-link=William H. Kautz |title=Optimized Data Encoding for Digital Computers |date=June 1954 |journal=Convention Record of the I.R.E. 1954 National Convention, Part 4: Electronic Computers and Information Technology |publisher=[[The Institute of Radio Engineers, Inc.]] |id=Session 19: Information Theory III - Speed and Computation |volume=2 |location=Stanford Research Institute, Stanford, California, USA |pages=47–57 |url=https://www.worldradiohistory.com/Archive-IRE/50s/IRE-1954-Part-4-Electronic-Computers-&-Information%20pdf |access-date=2020-05-22 }} (11 pages)</ref>\n<ref name=\"Stibitz_1941\">{{cite web |title=Complex Computer |author-first=George Robert |author-last=Stibitz |author-link=George Robert Stibitz |id=Patent US2668661A |orig-year=1941-04-19 |date=1954-02-09 |url=https://patents.google.com/patent/US2668661?oq=US2668661 |access-date=2020-05-24 }} [https://web.archive.org/web/20200523235356/https://patentimages.storage.googleapis.com/60/dc/f7/69cd9496a44f6d/US2668661.pdf] (102 pages)</ref>\n<ref name=\"Stibitz_1957\">{{cite book |title=Mathematics and Computers |author-first1=George Robert |author-last1=Stibitz |author-link1=George Robert Stibitz |author-first2=Jules A. |author-last2=Larrivee |date=1957 |edition=1 |publisher=[[McGraw-Hill Book Company, Inc.]] |publication-place=New York, USA / Toronto, Canada / London, UK |location=Underhill, Vermont, USA |lccn=56-10331 |page=105}} (10+228 pages)</ref>"}},"i":0}}]}'>
Richards, Richard Kohler (1955).
Arithmetic Operations in Digital Computers. New York, USA:
van Nostrand. p.
182.
↑ 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.)
↑ 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.
↑ Hayes, John P. (1978). Computer Architecture and Organization. McGraw-Hill International Book Company. p. 156. ISBN 0-07-027363-4.
↑