| Stibitz code | |
| Digits: | 4 |
| Tracks: | 4 |
| Digit Values: | 8 4 |
| Weight: | 1..3 |
| Continuity: | No |
| Cyclic: | No |
| Minimum Distance: | 1 |
| Maximum Distance: | 4 |
| Redundancy: | 0.7 |
| Lexicography: | 1 |
| Complement: | 9 |
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.
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):
| Decimal | Excess-3 | Stibitz | BCD 8-4-2-1 | Binary | 3-of-6 CCITT extension | 4-of-8 Hamming extension |
|---|---|---|---|---|---|---|
| -3 | 0000 | pseudo-tetrade | N/A | N/A | N/A | N/A |
| -2 | 0001 | pseudo-tetrade | ||||
| -1 | 0010 | pseudo-tetrade | ||||
| 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 | ||
| 12 | 1111 | pseudo-tetrade | pseudo-tetrade | 1100 | ||
| 13 | N/A | N/A | pseudo-tetrade | 1101 | ||
| 14 | pseudo-tetrade | 1110 | ||||
| 15 | pseudo-tetrade | 1111 |
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.)[1]
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.
BCD 8-4-2-1 to excess-3 converter example in VHDL:
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 isbegin 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
| 3-of-6 extension | |
| Digits: | 6 |
| Tracks: | 6 |
| Digit Values: | N/A --> |
| Weight: | 3 |
| Continuity: | No |
| Cyclic: | No |
| Minimum Distance: | 2 |
| Maximum Distance: | 6 |
| Hamming Distance: | --> |
| Redundancy: | --> |
| Lexicography: | 1 |
| Complement: | (9) |
| 4-of-8 extension | |
| Digits: | 8 |
| Tracks: | 8 |
| Digit Values: | N/A --> |
| Weight: | 4 |
| Continuity: | No |
| Cyclic: | No |
| Minimum Distance: | 4 |
| Maximum Distance: | 8 |
| Hamming Distance: | --> |
| Redundancy: | --> |
| Lexicography: | 1 |
| Complement: | 9 |