Algebraic and Stochastic Coding Theory

9

Matrix Codes

Since most ECCs are matrix codes, we present their common features about encoding and decoding and discuss certain special codes, which will be useful later in defining other important codes.

9.1    Matrix Group Codes

We use modulo 2 arithmetic. Let $F_2^n$ denote the binary n-tuples of the form a = (a₁, a₂, …, a_n) : a_i = 0 or 1 for i = 1, …, n. Then the properties of matrix codes are:

1.  An (m, n) code K is a one-to-one function $K:X\subset F_2^m\mapsto F_2^n,n\ge m.$

2.  The Hamming distance d (a, b) between a, b in $F_2^n$ is defined by

$\begin{array}{c}d(a,b)={\displaystyle \sum _{i=1}^n(a_i+b_i)}\\ =\text{number of coordinates for which }{a}_i\text{ and }{b}_i\text{ are different}\text{.}\end{array}$

3.  A code $K:F_2^m\mapsto F_2^n$ is a matrix code if x = x’A, where x’ denotes the (unique) complement of x and A is an m × n matrix.

4.  The range of the code ...