Binary Codes

The usual way of expressing a decimal number in terms of a binary number is known as pure binary coding and is discussed in the Number Systems section. A number of other techniques can be used to represent a decimal number. These are summarised below.

8421 BCD Code

In the 8421 Binary Coded Decimal (BCD) representation each decimal digit is converted to its 4-bit pure binary equivalent.

For example: 57_dec = 0101 0111_bcd

Addition is analogous to decimal addition with normal binary addition taking place from right to left. For example,

Where the result of any addition exceeds 9(1001) then six (0110) must be added to the sum to account for the six invalid BCD codes that are available with a 4-bit number. This is illustrated in the example below

Note that in the last example the 1 that carried forward from the first group of 4 bits has made a new 4-bit number and so represents the "1" in "15".

In the examples above the BCD numbers are split at every 4-bit boundary to make reading them easier. This is not necessary when writing a BCD number down.

This coding is an example of a binary coded (each decimal number maps to four bits) weighted (each bit represents a number, 1, 2, 4, etc.) code.

4221 BCD Code

The 4221 BCD code is another binary coded decimal code where each bit is weighted by 4, 2, 2 and 1 respectively. Unlike BCD coding there are no invalid representations. The decimal numbers 0 to 9 have the following 4221 equivalents

the 1's complement of a 4221 representation is important in decimal arithmetic. In forming the code remember the following rules

• Below decimal 5 use the right-most bit representing 2 first
• Above decimal 5 use the left-most bit representing 2 first
• Decimal 5 = 2+2+1 and not 4+1

Gray Code

Gray coding is an important code and is used for its speed, it is also relatively free from errors. In pure binary coding or 8421 BCD then counting from 7 (0111) to 8 (1000) requires 4 bits to be changed simultaneously. If this does not happen then various numbers could be momentarily generated during the transition so creating spurious numbers which could be read.

Gray coding avoids this since only one bit changes between subsequent numbers. To construct the code there are two simple rules. First start with all 0s and then proceed by changing the least significant bit (lsb) which will bring about a new state.

The first 16 Gray coded numbers are indicated below.

To convert a Gray-coded number to binary then follow this method

1. The binary number and the Gray-coded number will have the same number of bits
2. The binary MSB (left-hand bit) and Gray code MSB will always be the same
3. To get the binary next-to-MSB (i.e. next digit to the right) add the binary MSB and the gray code next-to-MSB. Record the sum, ignoring any carry.
4. Continue in this manner right through to the end.

Gray coding is a non-BCD, non-weighted reflected binary code.