New lower bounds on aperiodic crosscorrelation of binary codes

Abstract

For the minimum aperiodic crosscorrelation /spl theta/(n,M) of binary codes of size M and length n over the alphabet {1,-1} it is known that the celebrated Welch (1974) bound /spl theta//sup 2/(n,M)/spl ges/((M-1)n/sup 2/)/(2Mn-M-1). In this paper the Welch bound is strengthened for all M/spl ges/4 and n/spl ges/2. In the asymptotic process when M tends to infinity as n/spl rarr//spl infin/, this strengthening gives the factor 2 as compared to the Welch bound and coincides with the corresponding asymptotic bound on the square of the minimum periodic crosscorrelation of binary codes (Sidelnikov 1971). Our purpose is to estimate the aperiodic crosscorrelation /spl theta/(C) of a code C in E/sup n/={1,-1}/sup n/ which is defined as follows: /spl theta/(C)=max|/spl theta/(x,y;l)| where the maximum is taken over all x=(x/sub 1/,...,x/sub n/)/spl isin/C, y=(y/sub 1/,...,y/sub n/)/spl isin/C, l=0,l,...n-1 such that l/spl ne/0 when x=y and /spl theta/(x,y;l)=/spl Sigma//sub j=1//sup n-l/x/sub j/y/sub j/+l.