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.

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).

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.)^[14]

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] [15] [11]}

BCD 8-4-2-1 to excess-3 converter example in VHDL:

entitybcd8421xs3is
port(
a:instd_logic;
b:instd_logic;
c:instd_logic;
d:instd_logic;
an:bufferstd_logic;
bn:bufferstd_logic;
cn:bufferstd_logic;
dn:bufferstd_logic;
w:outstd_logic;
x:outstd_logic;
y:outstd_logic;
z:outstd_logic
);
endentitybcd8421xs3;
architecturedataflowofbcd8421xs3is
begin
an<=nota;
bn<=notb;
cn<=notc;
dn<=notd;
w<=(anandbandd)or(aandbnandcn)
or(anandbandcanddn);
x<=(anandbnandd)or(anandbnandcanddn)
or(anandbandcnanddn)or(aandbnandcnandd);
y<=(anandcnanddn)or(anandcandd)
or(aandbnandcnanddn);
z<=(ananddn)or(aandbnandcnanddn);
endarchitecturedataflow;-- of bcd8421xs3

• 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 ^[16] (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]

• Offset binary, excess-N, biased representation
• Excess-128
• Excess-Gray code
• Shifted Gray code
• Gray code
• m-of-n code
• Aiken code

• Binary arithmetic
• Numeral systems

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