Excess-3

Excess-3, 3-excess or 10-excess-3 binary code (often abbreviated as XS-3, 3XS or X3), shifted binary or Stibitz code (after George Stibitz, who built a relay-based adding machine in 1937) 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.
The greatest binary number represents the largest value.
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.)
Motivation
The primary advantage of excess-3 coding over non-biased coding is that a decimal number can be nines' complemented (for subtraction) as easily as a binary number can be ones' complemented: just by inverting all bits. 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.
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 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.
4-of-8 code extension: As an alternative to the IBM transceiver code (which is a 4-of-8 code with a Hamming distance of 2), 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.
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
Binary arithmetic
Numeral systems