Introduction to Coding Theory

Abstract

A binary block code is a set of n-dimensional vectors over the binary field. Each of these vectors is called a codeword. All n-dimensional vectors, including those not in the code, are called words. The number of components in each word, n, is called the block length. The code is said to be linear iff the sum of every pair of codewords is another codeword. Of course, the sum of two binary vectors is computed component by component without carries, according to the rules 0 + 0 = 1 + 1 = 0, 0 + 1 = 1 + 0 = 1. The number of codewords in a linear code is a power of 2, say 2^k and the code may be conveniently specified as the row space of a k×n binary generator matrix, G whose rows may be taken as any k linearly independent codewords. Alternatively, the code may also be defined as the null-space of a (n−k)×n binary parity-check matrix, H which satisfies

$$$$

The row space of H is the orthogonal complement of the row space of G.