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A strengthening of the Assmus-Mattson theorem

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3 Author(s)
Calderbank, A.R. ; AT&T Bell Lab., Murray Hill, NJ, USA ; Delsarte, P. ; Sloane, N.

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'3, be the nonzero weights in the dual code C1. 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'in-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 w2d+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

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Information Theory, IEEE Transactions on  (Volume:37 ,  Issue: 5 )