Excess-3

"XS-3" redirects here. For the experimental aircraft, see Douglas XS-3 Stiletto.

"Shifted binary" redirects here. For the general concept, see Offset binary. For binary shifts, see Bit shifting.

Stibitz code
Digits	4 [1]
Tracks	4 [1]
Digit values	8  4 −2 −1
Weight(s)	1..3 [1]
Continuity	No [1]
Cyclic	No [1]
Minimum distance	1 [1]
Maximum distance	4
Redundancy	0.7
Lexicography	1 [1]
Complement	9 [1]
v t e

Excess-3, 3-excess ^{[1] [2] [3]} or 10-excess-3 binary code (often abbreviated as XS-3, ^[4] 3XS ^[1] or X3 ^{[5] [6]} ), shifted binary ^[7] or Stibitz code ^{[1] [2] [8] [9]} (after George Stibitz, ^[10] who built a relay-based adding machine in 1937 ^{[11] [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

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 (2^N+1 − excess − 1).

Excess-3, and Stibitz code

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.) ^[14]

Advantage

The primary advantage of excess-3 coding over non-biased coding is that a decimal number can be nines' complemented ^[1] (for subtraction) as easily as a binary number can be ones' complemented: just by inverting all bits. ^[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. ^{[1] [15] [11]}

Example

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

entitybcd8421xs3isport(a:instd_logic;b:instd_logic;c:instd_logic;d:instd_logic;an:bufferstd_logic;bn:bufferstd_logic;cn:bufferstd_logic;dn:bufferstd_logic;w:outstd_logic;x:outstd_logic;y:outstd_logic;z:outstd_logic);endentitybcd8421xs3;architecturedataflowofbcd8421xs3isbeginan<=nota;bn<=notb;cn<=notc;dn<=notd;w<=(anandbandd)or(aandbnandcn)or(anandbandcanddn);x<=(anandbnandd)or(anandbnandcanddn)or(anandbandcnanddn)or(aandbnandcnandd);y<=(anandcnanddn)or(anandcandd)or(aandbnandcnanddn);z<=(ananddn)or(aandbnandcnanddn);endarchitecturedataflow;-- of bcd8421xs3

Extensions

3-of-6 extension
Digits	6 [1]
Tracks	6 [1]
Weight(s)	3 [1]
Continuity	No [1]
Cyclic	No [1]
Minimum distance	2 [1]
Maximum distance	6
Lexicography	1 [1]
Complement	(9) [1]
v t e

4-of-8 extension
Digits	8 [1]
Tracks	8 [1]
Weight(s)	4 [1]
Continuity	No [1]
Cyclic	No [1]
Minimum distance	4 [1]
Maximum distance	8
Lexicography	1 [1]
Complement	9 [1]
v t e

• 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. ^{[13] [1]}
• 4-of-8 code extension: As an alternative to the IBM transceiver code ^[16] (which is a 4-of-8 code with a Hamming distance of 2), ^[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. ^[1]

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.
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. Richards, Richard Kohler (1955). Arithmetic Operations in Digital Computers. New York, USA: van Nostrand. p. 182.
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)
5. 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.
6. 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.)
7. 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)
8. 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.)
9. 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.)
10. Stibitz, George Robert (1954-02-09) [1941-04-19]. "Complex Computer". Patent US2668661A. Retrieved 2020-05-24. (102 pages)
11. 1 2 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.
12. Ritchie, David (1986). The Computer Pioneers. New York, USA: Simon and Schuster. p.  35. ISBN   067152397X.
13. Comité Consultatif International Téléphonique et Télégraphique (CCITT), Groupe de Travail 43 (1959-06-03). Contribution No. 1. CCITT, GT 43 No. 1.
14. Hayes, John P. (1978). Computer Architecture and Organization. McGraw-Hill International Book Company. p. 156. ISBN   0-07-027363-4.
15. 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.
16. IBM (July 1957). 65 Data Transceiver / 66 Printing Data Receiver.