Math @ Duke

Publications [#235802] of Robert Calderbank
Papers Published
 Calderbank, AR; Delsarte, P; Sloane, NJA, A strengthening of the AssmusMattson theorem,
IEEE Transactions on Information Theory, vol. 37 no. 5
(1991),
pp. 12611268 [doi]
(last updated on 2018/02/21)
Abstract: Let w1 = d,w2,...,ws be 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 C⊥. Let t be an integer in the range 0 < t < d such that there are at most d  t weights w′i with 0 < w′i ≤ n  t. E. F. Assmus and H. F. Mattson, Jr. (1969) proved that the words of any weight wi in C form a tdesign. The authors 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 selfdual with all weights divisible by 4 then the codewords of any given weight wi form either a (t + 1)design or a {1,2,...,t,t + 2}design. The proof avoids the use of modular forms.


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