## A strengthening of the Assmus-Mattson Theorem

Summary form only given. Let w1 = d, w2, ..., ws be the weights of the nonzero code words in a binary linear [n, k, d] code C, and let w1′, w2′, ..., ws′ 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 wi′ with 0 < wi′ ≤ n - t. Assmus and Mattson proved that the words of any weight wi in C form a t-design. Let δ = 0 or 1, according to whether C is even or not, and let B denote the set of code words of weight d. The present authors have proved that if w2 ≥ d + 4, then either (1) t = 1, d is odd, and B partitions {1, 2, ..., n}, or (2) B is a (t + δ + 1)-design, or (3) B is a {1, ..., t + δ, t + δ + 2}-design. If C is a self-orthogonal binary code with all weights divisible by 4, then the result extends to code words of any given weight. The special case of code words of minimal weight in extremal self-dual codes also follows from a theorem of Venkov and Koch.

### Duke Scholars

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*A strengthening of the Assmus-Mattson Theorem*. Dec. 1990, p. 41.