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).
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
- Offset binary, excess-N, biased representation
- Excess-128
- Excess-Gray code
- Shifted Gray code
- Gray code
- m-of-n code
- Aiken code
References
- ^ a b c d e f g h i j k l m n o p q r s t u v w x y z aa ab ac ad ae af ag ah ai 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.
- ^ a b 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) - ^ Richards, Richard Kohler (1955). Arithmetic Operations in Digital Computers. New York, USA: van Nostrand. p. 182.
- ^ 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.
- ^ 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.)
- ^ a b 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.
{{cite web}}
: Unknown parameter|dead-url=
ignored (|url-status=
suggested) (help) - ^ Ritchie, David (1986). The Computer Pioneers. New York, USA: Simon and Schuster. p. 35. ISBN 067152397X.
- ^ a b Comité 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) - ^ 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.
- ^ IBM (July 1957). 65 Data Transceiver / 66 Printing Data Receiver.