# 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
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]