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 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 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. 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 and 2, the sum looks like 6 instead of 3. In order to correct this problem, after adding two digits, it is necessary to remove the extra bias by subtracting binary 0011 if the resulting digit is less than decimal 10, or subtracting binary 1101 if an overflow has occurred.
Motivation
The primary advantage of excess-3 coding over non-biased coding is that a decimal number can be nines' complemented 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. 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
to excess-3 converter example in VHDL: entity bcd8421xs3 is port ; end entity bcd8421xs3; architecture dataflow of bcd8421xs3 is begin an <= not a; bn <= not b; cn <= not c; dn <= not d; w <= or or ; x <= or or or ; y <= or or ; z <= or ; 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 IBMtransceiver code, 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.
