Excess-3

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.

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]

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]
−3	0000	pseudo-tetrade	N/A	N/A	N/A	N/A
−2	0001	pseudo-tetrade
−1	0010	pseudo-tetrade
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
10	1101	pseudo-tetrade	pseudo-tetrade	1010	N/A	N/A
11	1110	pseudo-tetrade	pseudo-tetrade	1011
12	1111	pseudo-tetrade	pseudo-tetrade	1100
13	N/A	N/A	pseudo-tetrade	1101
14			pseudo-tetrade	1110
15			pseudo-tetrade	1111

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). In order 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.)

Motivation

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][14][11]

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 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]

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]

• 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[15] (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. Taschenbuch der Nachrichtenverarbeitung (in German). 2 (3 ed.). Berlin, Germany: Springer Verlag. pp. 98–100. ISBN 3-540-06241-6. LCCN 73-80607.
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. Stanford Research Institute, Stanford, California, USA: The Institute of Radio Engineers, Inc. 2: 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. (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). 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. 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. 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.
15. IBM (July 1957). 65 Data Transceiver / 66 Printing Data Receiver.