Excess-3

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]

Excess-3, 3-excess[1][2][3] or 10-excess-3 binary code (often abbreviated as XS-3, 3XS[1] or X3[4][5]) or Stibitz code[1][2] (after George Stibitz, who built a relay-based adding machine in 1937[6][7]) 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 (2N+1 − excess − 1).
Excess-3 / Stibitz code
Decimal Excess-3 Stibitz BCD 8-4-2-1 Binary 3-of-6 CCITT extension[8][1] 4-of-8 Hamming extension[1]
−3 0000 pseudo-tetrade N/A N/A N/A N/A
−2 0001 pseudo-tetrade N/A N/A N/A N/A
−1 0010 pseudo-tetrade N/A N/A N/A N/A
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 N/A N/A
12 1111 pseudo-tetrade pseudo-tetrade 1100 N/A N/A
13 N/A N/A pseudo-tetrade 1101 N/A N/A
14 N/A N/A pseudo-tetrade 1110 N/A N/A
15 N/A N/A pseudo-tetrade 1111 N/A N/A

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][9][6]

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  : inout std_logic;
    bn  : inout std_logic;
    cn  : inout std_logic;
    dn  : inout 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.[8][1]
  • 4-of-8 code extension: As an alternative to the IBM transceiver code[10] (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

References

  1. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 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. 1 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. ↑ 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.
  5. ↑ 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.)
  6. 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.
  7. ↑ Ritchie, David (1986). The Computer Pioneers. New York, USA: Simon and Schuster. p. 35. ISBN 067152397X.
  8. 1 2 Comité Consultatif International Téléphonique et Télégraphique (CCITT), Groupe de Travail 43 (1959-06-03). Contribution No. 1. CCITT, GT 43 No. 1.
  9. ↑ Bashe, C. J.; Jackson, P. W.; Mussell, H. A.; Winger, W. D. (January 1956). "The Design of the IBM Type 702 System". Transactions of the American Institute of Electrical Engineers (AIEE), Part 1: Communication and Electronics. 74 (6): 695–704. doi:10.1109/TCE.1956.6372444. Paper No. 55-719.
  10. ↑ IBM (July 1957). 65 Data Transceiver / 66 Printing Data Receiver.
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