Search

In recent years a number of nonlinear codes have been discovered which have better error-correcting capabilities than any known linear codes. However, very little is known about the properties of such codes. In this paper we study the most basic property, the weight enumerator. The weight of a codeword is the number of its nonzero components; the weight enumerator gives the number of codewords of each weight, and is fundamental for obtaining the error probability when the code is used for error-correction on a noisy channel. In 1963 one of us showed that the weight enumerator of a linear code is related in a simple way to that of the dual code (Jessie MacWilliams, “A Theorem on the Distribution of Weights in a Systematic Code,” Bell System Technical Journal, 42, No. 1 (January 1963), pp. 79–94). In the present paper, which is a sequel, we show that the same relationship holds for the weight enumerator of a nonlinear code. Furthermore, a definition is given for the dual α⊥ of a nonlinear binary code α which satisfies (α⊥)⊥ = α provided α contains the zero codeword.

A class of binary codes, satisfying the equality in a specialized version of the Johnson bound, is introduced, which contains perfect codes and the Preparata 2-error correcting codes. These codes are shown to contain t-designs, which can be extended to (t + 1)-designs. It is shown how the weight-distribution of these codes can be uniquely derived.

A subclass of cyclic codes of composite block length n=n' \cdot n" , with n' and n" relatively prime, is studied. It is shown how, under certain conditions, these codes may be factorized into subcodes of block length n' and n" and how they are related to codes of block length n' or n" , over extension fields of the base field. The problem is, in a sense, the converse of the one considered by Burton and Weldon, [5] and some of our codes are subcodes of their product codes. The factorization property gives better insight into the structure of various codes and allows, in particular, to obtain new lower bounds on their minimum weight, hence improving in many cases on the Bose-Chaudhuri bound.

Hadamard matrices exist for infinitely many orders 4m, m ≧ 1, m integer, including all 4m < 100, cf. [3], [2]. They are conjectured to exist for all such orders. Skew Hadamard matrices have been constructed for all orders 4m < 100 except for 36, 52, 76, 92, cf. the table in [4]. Recently Szekeres [6] found skew Hadamard matrices of the order 2(pt +1)≡ 12 (mod 16), p prime, thus covering the case 76. In addition, Blatt and Szekeres [1] constructed one of order 52. The present note contains a skew Hadamard matrix of order 36 (and one of order 52), thus leaving 92 as the smallest open case.