Department of Mathematics
 Search | Help | Login | pdf version | printable version

Math @ Duke



Publications [#236030] of Robert Calderbank

Papers Published

  1. Calderbank, AR; Delsarte, P; Sloane, NJA, A strengthening of the Assmus-Mattson Theorem (1990), pp. 41-
    (last updated on 2017/12/15)

    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.
ph: 919.660.2800
fax: 919.660.2821

Mathematics Department
Duke University, Box 90320
Durham, NC 27708-0320