New lower bounds on aperiodic crosscorrelation of binary codes

For the minimum aperiodic crosscorrelation θ(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 θ²(n,M)⩾((M-1)n ²)/(2Mn-M-1). In this paper the Welch bound is strengthened for all M⩾4 and n⩾2. In the asymptotic process when M tends to infinity as n→∞, 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 θ(C) of a code C in Eⁿ={1,-1}ⁿ which is defined as follows: θ(C)=max|θ(x,y;l)| where the maximum is taken over all x=(x₁,...,x_n)∈C, y=(y₁,...,y_n )∈C, l=0,l,...n-1 such that l≠0 when x=y and θ(x,y;l)=Σ_j=1^n-lx_jy_j +l