Binomial ideal associated to a lattice and its label code

Abstract

In coding theory the study of the binomial ideal derived from an arbitrary code is currently of great interest; see for example [5]. This is mainly because of a known relation between binomial ideals and lattices or codes. Also, studying the relation between binomial ideals associated to a lattice and its label code helps to solve the closest vector problem in lattices as well as decoding binary and non-binary codes [1, 3] and finding a label code of a lattice, as we do in this work. Every lattice Λ can be described in terms of a label code L and an orthogonal sublattice Λ′ such that Λ/Λ′ ∼= L [2]. We assign binomial ideals IΛ and IL to an integer lattice Λ and its label code L, respectively. In this work, we identify the binomial ideal associated to an integer lattice and then establish the relation IΛ = IΛ′ + IL between the ideal of the lattice and its label code. In this work, we define a binomial ideal for an integer lattice and its label code slightly different from [1, 3, 4, 7]. Let K[X] = K[x1, . . . , xn] denote the polynomial ring, where K is an arbitrary field. Consider ≺ as a fixed total degree compatible term order with x1 x2 · · · xn. The monomials in K[X] are denoted by X = x1 1 . . . x bn n where b = (b1, . . . , bn) is an element of N0 and N0 is the set of non-negative integers. We use the notation