69,432 research outputs found
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 such that C is a
punctured code of the kernel ker of the incidence matrix of
over F and there is a linear mapping between C and ker 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 and the Dobbertin
APN function . 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 defined by ,
where is a primitive element in . 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
Construction of minimal linear codes with few weights from weakly regular plateaued functions
The construction of linear (minimal) codes from functions over finite fields has been greatly studied in the literature since determining the parameters of linear codes based on functions is rather easy due to the nice structures of functions. In this paper, we derive 3-weight and 4-weight linear codes from weakly regular plateaued unbalanced functions in the recent construction method of linear codes over the odd characteristic finite fields. The Hamming weights and their weight distributions for proposed codes are determined by using the Walsh transform values and Walsh distribution of the employed functions, respectively. We next derive projective 3-weight punctured codes with good parameters from the constructed codes. These punctured codes may be almost optimal due to the Griesmer bound, and they can be employed to design association schemes. We lastly show that all constructed codes are minimal, which approves that they can be employed to design high democratic secret sharing schemes
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
- …