On the notion of Jacobi polynomials for codes

Abstract

In this paper, we introduce the notion of Jacobi polynomials for codes, and establish some fundamental features of it. This notion comes out of considerations on the various invariants of the codes (e.g. [22–24]); it has to do with Jacobi theta-series of positive definite quadratic forms (cf. [9], p. 82). Thus the name ‘Jacobi polynomials for codes’ traces this fact.

A Study of Deep Holes in the First-Order Reed-Muller Codes

“…We regard these binary vectors as the characteristic vectors for the 2 elements subsets in an eight elements set. We naturally arrive at a Johnson scheme J (8,2). This scheme consists of all the 2-elements subsets in an eight elements set X.…”

“…Two elements subsets x and y of X are connected by the hth relation R h (0 ≤ h ≤ 2) if and only if the condition |x ∩ y| = 2 − h holds. To each deep hole v we correspond the 2 elements setv consisting of non-zero coordinate positions of v. It can be observed that the mapping v →v is a bijection between the set of all deep holes for RM (1,3) and J (8,2). Further it holds that…”

“…We note that the first-order Reed-Muller code RM (1,3) is equivalent to the extended binary Hamming [8,4,4] code of length 8, and this code is known to have the covering radius 2 and all the vectors of weight 2 are the coset leaders for the code RM (1,3). This implies that these coset leaders form a Johnson scheme J (8,2).…”

“…This implies that these coset leaders form a Johnson scheme J (8,2). As to the definitions of the Hamming association scheme and the Johnson scheme one may refer to Delsarte [3,4] or Bannai and Ito [1].…”

Modular invariance property of association schemes, type II codes over finite rings and finite abelian groups and reminiscences of François Jaeger (a survey)

“…Ozeki [42] or Runge [43]) that if we take the index 2 subgroup H (of order 96) of G defined by The important implication of this fact is that we can understand the space of modular forms completely through the invariant ring of the finite group H. Interestingly enough, this situation can be generalized in several directions. We list some of them in the following table.…”