Calderbank, AR; Delsarte, P; Sloane, NJA, *A strengthening of the Assmus-Mattson theorem*,
IEEE Transactions on Information Theory, vol. 37 no. 5
(1991),
pp. 1261-1268 [doi] .
**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 t-design. 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 self-dual 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.*