Geometric representations of linear codes

We say that a linear code C over a field F is triangular representable if there exists a two dimensional simplicial complex $\Delta$ such that C is a punctured code of the kernel ker $\Delta$ of the incidence matrix of $\Delta$ over F and there is a linear mapping between C and ker $\Delta$ which is a bijection and maps minimal codewords to minimal codewords. We show that the linear codes over rationals and over GF(p), where p is a prime, are triangular representable. In the case of finite fields, we show that this representation determines the weight enumerator of C. We present one application of this result to the partition function of the Potts model. On the other hand, we show that there exist linear codes over any field different from rationals and GF(p), p prime, that are not triangular representable. We show that every construction of triangular representation fails on a very weak condition that a linear code and its triangular representation have to have the same dimension.Comment: 20 pages, 8 figures, v3 major change

A Note on Cyclic Codes from APN Functions

Cyclic codes, as linear block error-correcting codes in coding theory, play a vital role and have wide applications. Ding in \cite{D} constructed a number of classes of cyclic codes from almost perfect nonlinear (APN) functions and planar functions over finite fields and presented ten open problems on cyclic codes from highly nonlinear functions. In this paper, we consider two open problems involving the inverse APN functions $f(x)=x^{q^m-2}$ and the Dobbertin APN function $f(x)=x^{2^{4i}+2^{3i}+2^{2i}+2^{i}-1}$. From the calculation of linear spans and the minimal polynomials of two sequences generated by these two classes of APN functions, the dimensions of the corresponding cyclic codes are determined and lower bounds on the minimum weight of these cyclic codes are presented. Actually, we present a framework for the minimal polynomial and linear span of the sequence $s^{\infty}$ defined by $s_t=Tr((1+\alpha^t)^e)$, where $\alpha$ is a primitive element in $GF(q)$. These techniques can also be applied into other open problems in \cite{D}

The weight hierarchies and chain condition of a class of codes from varieties over finite fields

The generalized Hamming weights of linear codes were first introduced by Wei. These are fundamental parameters related to the minimal overlap structures of the subcodes and very useful in several fields. It was found that the chain condition of a linear code is convenient in studying the generalized Hamming weights of the product codes. In this paper we consider a class of codes defined over some varieties in projective spaces over finite fields, whose generalized Hamming weights can be determined by studying the orbits of subspaces of the projective spaces under the actions of classical groups over finite fields, i.e., the symplectic groups, the unitary groups and orthogonal groups. We give the weight hierarchies and generalized weight spectra of the codes from Hermitian varieties and prove that the codes satisfy the chain condition

On the Hamming distance of linear codes over a finite chain ring

Let R be a finite chain ring (e.g. a Galois ring), K its residue field and C a linear code over R. We prove that d(C), the Hamming distance of C, is d((C : α)), where (C : α) is a submodule quotient, α is a certain element of R and — denotes the canonical projection to K. These two codes also have the same set of minimal codeword supports. We explicitly construct a generator matrix/polynomial of (C : α) from the generator matrix/polynomials of C. We show that in general d(C) ≤ d(C) with equality for free codes (i.e. for free R- submodules of Rn) and in particular for Hensel lifts of cyclic codes over K. Most of the codes over rings described in the literature fall into this class. We characterise MDS codes over R and prove several analogues of properties of MDS codes over finite fields. We compute the Hamming weight enumerator of a free MDS code over R