Multiplying Vectors in Binary Quadratic Residue Codes
Robert Calderbank, David B. Wales · SIAM Journal on Algebraic and Discrete Methods · 1982
Let $q \equiv 1 (\bmod 8)$ be an odd prime power and let $C( q )$ and $C ( q )^*$ be the two extended binary quadratic residue codes of length $( q + 1 )$. We show how to regard $C( q )$ and $C( q )^*$ as one-sided ideals in a binary group algebra and we show that the appropriate product is a 2-dimensional ideal. This allows us to prove that if d is the minimum weight in $C ( q )$ then $( d - 1 )^2 - ( d - 1 ) + 1 - st\geqq q$, where s, t are nonnegative integers, with $s \equiv 0 (\bmod 4)$, and t odd. The integers s and t depend on the way the nonzero entries of a codeword of minimum weight are distributed among the coordinate positions. We prove that $( d - 1 )^2 - ( d - 1 ) + 1 = q$ only if $q = 7$ and $d = 4$. We also investigate the case $s = 0$.