New lower bounds on aperiodic crosscorrelation of binary codes

Abstract:

For the minimum aperiodic crosscorrelation θ (n,M) of binary codes of size M and length n over the alphabet {1,-1} there exists the celebrated Welch bound which was published in 1974 and has remained in this form up to now. In the paper this bound is strengthened for all
M ≥ 4 and n ≥ 2. The main idea of the proof is a new sufficient condition for the mean value of a nonnegative definite matrix over the code to be greater than or equal to the average over the whole space. This allows one to take into account weights of cyclic shifts of code vectors and solve the problem of their optimal choice.

Keywords:

binary codes, aperiodic crosscorrelation, Welch bound, inequality on the mean, quadratic forms, Chebyshev polynomials.

Publication language:english,
pages:4

Research direction:

Mathematical problems and theory of numerical methods