Weights of Codewords in Random Binary Linear Codes and Their Correlation Properties

A classical problem in coding theory addresses moments of the weight spectrum distribution. The results in this work are on the weight moments of individual codewords rather than the weight spectrum. The expectations of single and pairwise products of weights of nonzero words in a random binary linear code are given. We show that the covariance between the weights of any pair of distinct nonzero words is zero. Our main theorem has an application to sequence correlations problem. We prove that the sums of out of phase self correlations, as well as sums of cross-correlations, of nonzero words in a random binary linear code are equal to zero.