Digital codes

Computers and electronic devices need a systematic, precise way to represent information so that every character, letter, digit and symbol is unique and distinguishable. Analog data is therefore processed into digital data that is represented, stored and transmitted as groups of binary bits. A digital code is simply a rule for converting information into such groups of 0s and 1s.

Codes are broadly of three kinds: a binary (digital) code is a group of bits using only 0 and 1; a numeric code represents numbers; and an alphanumeric code represents letters, characters and numerals. Numeric codes are further classified as weighted codes, which obey the positional-weight principle (each bit position carries a fixed weight, e.g. 8-4-2-1, 2-4-2-1, 5-4-2-1), and non-weighted codes, which do not (e.g. Excess-3 and Gray code).

BCD (8-4-2-1) code

In Binary Coded Decimal (BCD) each decimal digit is represented by its own group of four bits. The most common is the 8-4-2-1 BCD code, where the four bit positions carry weights 8, 4, 2 and 1 from left to right (table 10.2.1). Because four bits can form sixteen combinations but only ten are needed, six combinations (1010 to 1111) are invalid in BCD. BCD is less efficient than pure binary but is easy to convert to and from decimal, so it is used in calculators, seven-segment displays and digital watches. For a multi-digit number each decimal digit is coded separately — for example, decimal 42 becomes 0100 0010.

Self-complementing codes and 9's complement

A code is self-complementing when the code of the 9's-complement of a digit is obtained simply by interchanging all the 1s and 0s (taking the 1's complement) of that digit's code. The 9's complement of a decimal digit is found by subtracting it from 9 (table 10.2.2); for a multi-digit number each digit is complemented separately — e.g. the 9's complement of 7825 is 2174.

Excess-3 (XS-3) code

The Excess-3 code is a non-weighted, self-complementing code formed by adding 0011 (binary) — that is, 3 (decimal) — to each 8-4-2-1 BCD code word (table 10.2.3). Its self-complementing property makes it convenient for arithmetic subtraction. Six of its sixteen combinations (0000, 0001, 0010, 1101, 1110, 1111) are invalid.

2-4-2-1 code

The 2-4-2-1 code is a weighted, self-complementing 4-bit code whose bit weights from left to right are 2, 4, 2 and 1 (table 10.2.4). Because two positions share the weight 2, some digits could be coded in more than one way; to preserve the self-complementing property the standard coding is used and six combinations (0101, 0110, 0111, 1000, 1001, 1010) must not be used. It is widely used in arithmetic operations.

Gray code

The Gray code is a non-weighted unit-distance (minimum-change) code: successive numbers differ by only one bit (table 10.2.5). Because it is non-weighted it cannot be used directly for arithmetic, but the single-bit change makes it valuable in shaft-position encoders, analog-to-digital conversion and reducing errors in data transmission.

• Binary → Gray: copy the MSB unchanged, then each remaining Gray bit is the XOR of two adjacent binary bits. Figure 10.2.1 realizes this with three XOR gates.
• Gray → Binary: copy the MSB unchanged, then each binary bit is the XOR of the previous binary bit with the current Gray bit. Figure 10.2.2 realizes this with three XOR gates.

Alphanumeric codes

Alphanumeric codes represent digits, alphabets and special characters (at least 10 digits and 26 letters). The three common ones are ASCII, EBCDIC and the five-bit Baudot code. …