A Strengthening of the Assmus-mattson Theorem
(1991) IEEE Transactions on Information Theory — Vol. 37, n° 5, p. 1261-1268 (1991)
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- Let w1 = d,w2,...,w(s) 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 perpendicular-to. Let t be an integer in the range 0 < t < d such that there are at most d - t weights w'(t) with 0 < w'(i) less-than-or-equal-to n - t. Assumus and Mattson proved that the words of any weight w(i) in C form a t-design. We show that if w2 greater-than-or-equal-to d + 4 then either the words of any nonzero weight w(i) 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 w(i) form 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.
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Calderbank, AR., Delsarte, P., & Sloane, NJA. (1991). A Strengthening of the Assmus-mattson Theorem. IEEE Transactions on Information Theory, 37(5), 1261-1268. https://doi.org/10.1109/18.133244 (Original work published 1991)