A Strengthening of the Assmus-Mattson Theorem

Let W1 = d, w2,…, wsbe the weights of the nonzero codewords in a binary linear [n, k, d] code C, and let w'1, w'2,…, w's' be the nonzero weights in the dual code CT. Let t be an integer in the range 0 < t < d such that there are at most d — t weights w'iwith 0 < w'i ≤ n — t. Assmus and Mattson proved that the words of any weight wi in C form a t-design. We show that if w2≤d + 4 then either the words of any nonzero weight wi form a (t+1)-design or else the codewords of minimal weight d form a {1,2,…, t, t+2}-design. If in addition C is self-dual with all weights divisible by 4 then the codewords of any given weight wiform either a (t + 1)-design or a {1,2,…, t, t + 2}-design. The special case of this result for codewords of minimal weight in an extremal self-dual code with all weights divisible by 4 also follows from a theorem of Venkov and Koch; however our proof avoids the use of modular forms. © 1991 IEEE

Duke Scholars

Author Robert Calderbank Computer Science