On minimum weight codewords in QR codes
A linear (n,k,d) code is said to be optimal iff \(d=n-k+1\). Assmus and Mattson have shown that given a prime p, for all but a finite number of primes q each cyclic (p,k) code over \(GF(q^ i)\) is optimal. This paper provides a method for determining these ``exceptional'' characteristics, and the corresponding minimum weights, for quadratic-residue (QR) codes of modest length. The procedure involves calculations with global QR codes. The analysis reveals that a nonoptimal case requires \(q\leq 2^{- (p-1)/2}[(p+1)/2]^{(p+1)/4}.\) In addition, the method can be used to examine the t-designs derived from the codewords of minimum weight.
scientific article; zbMATH DE number 4049545