Optimal generator matrices of binary cyclic codes over the binary symmetric channel

This paper considers the optimal generator matrices of a binary cyclic code over a binary symmetric channel with crossover probability p rarr 0 when the goal is to minimize the probability of an information bit error. A given code has many encoder realizations and the information bit error probability is a function of this realization. We derive some sufficient conditions for a binary cyclic code to have systematic optimal generator matrices under bounded distance decoding and determine many binary cyclic codes satisfying these properties.