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.
We use modulo 2 arithmetic. Let 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
2. The Hamming distance d (a, b) between a, b in is defined by
3. A code 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 ...