TABLE OF CONTENTS Knowing a basis for a linear code enables us to describe its codewords explicitly. In coding theory, a basis for a linear code is often represented in the form of a matrix, called a generator matrix, while a matrix that represents a basis for the dual code is called a parity-check matrix. These matrices play an important role in coding theory.
(i) A generator matrix for a linear code C is a matrix G whose rows form a basis for C.
(ii) A parity-check matrix H for a linear code C is a generator matrix for the dual code C ?.
(i) If C is an [ n, k]-linear code, then a generator matrix for C must be a k n matrix and a parity-check matrix for C must be an ( n ? k) n matrix.
(ii) Algorithm 4.3 of Section 4.4 can be used to find generator and parity-check matrices for a linear code.
(iii) As the number of bases for a vector space usually exceeds one, the number of generator matrices for a linear code also usually exceeds one. Moreover, even when the basis is fixed, a permutation (different from the identity) of the rows of a generator matrix also leads to a different generator matrix.
(iv) The rows of a generator matrix are linearly independent. The same holds for the...
