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 F2n denote the binary n-tuples of the form a = (a1, a2, …, an) : ai = 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:XF2mF2n,nm.

2.  The Hamming distance d (a, b) between a, b in F2n is defined by

d(a,b)=i=1n(ai+bi)=number of coordinates for which ai and bi are different.

3.  A code K:F2mF2n is a matrix code if x = xA, where x’ denotes the (unique) complement of x and A is an m × n matrix.

4.  The range of the code ...

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