Relative Generalized Hamming Weights Research Articles

Relative generalized Hamming weights of evaluation codes

This work aims to algebraically describe the relative generalized Hamming weights of evaluation codes. We give a lower bound for these weights in terms of a footprint bound, and prove that this bound can be sharp. We compute the next-to-minimal weight of toric codes over hypersimplices of degree 1.

Further results on higher weights of codes over finite rings

The descriptions of generalized Hamming weights with respect to rank are given for codes over chain rings. Based on the descriptions of generalized Hamming weights with respect to rank, the double chain condition, in particular, the chain condition, is introduced and some judging criteria for the double chain condition are presented. As an application of the chain condition, we determine generalized Hamming weights with respect to rank of the tensor product of certain codes satisfying the chain condition. By using generalized Hamming weights with respect to rank, we generalize some results obtained in recent references. We introduce relative generalized Hamming weights to codes over chain rings and principal ideal rings, and present equivalent descriptions and bounds on them. We also generalize maximum distance codes with respect to rank to relative maximum distance codes with respect to rank and give a series of judging criteria for relative maximum distance codes.

Relative generalized Hamming weights over Frobenius rings

Relative generalized Hamming weights over Frobenius rings

Relative generalized Hamming weights of affine Cartesian codes

We explicitly determine all the relative generalized Hamming weights of affine Cartesian codes using the notion of footprints and results from extremal combinatorics. This generalizes the previous works on the determination of relative generalized Hamming weights of Reed–Muller codes by Geil and Martin, as well as the determination of all the generalized Hamming weights of the affine Cartesian codes by Beelen and Datta.

Relative profiles and extended weight polynomials of almost affine codes

In this paper we study various aspects concerning almost affine codes, a class including, and strictly larger than, that of linear codes. We use the combinatorial tool demi-matroids to show how one can define relative length/dimension and dimension/length profiles of flags (chains) of almost affine codes. In addition we show two specific results. One such result is how one can express the relative length/dimension profiles (also called relative generalized Hamming weights) of a pair of codes in terms of intersection properties between the smallest of these codes and subcodes of the largest code. The other result tells how one can find the extended weight polynomials, expressing the number of codewords of each possible weight, for each code in an infinite hierarchy of extensions of a code over a given alphabet.

Relative Generalized Matrix Weights of Matrix Codes for Universal Security on Wire-Tap Networks

Universal security over a network with linear network coding has been intensively studied. However, previous linear codes and code pairs used for this purpose were linear over a larger field than that used on the network, which restricts the possible packet lengths of optimal universal secure codes, does not allow to apply known list-decodable rank-metric codes and requires performing operations over a large field. In this paper, we introduce new parameters (relative generalized matrix weights and relative dimension/rank support profile) for code pairs that are linear over the field used in the network, and show that they measure the universal security performance of these code pairs. For one code and non-square matrices, generalized matrix weights coincide with the existing Delsarte generalized weights, hence we prove the connection between these latter weights and secure network coding, which was left open. As main applications, the proposed new parameters enable us to: 1) obtain optimal universal secure linear codes on noiseless networks for all possible packet lengths, in particular for packet lengths not considered before, 2) obtain the first universal secure list-decodable rank-metric code pairs with polynomial-sized lists, based on a recent construction by Guruswami et al; and 3) obtain new characterizations of security equivalences of linear codes. Finally, we show that our parameters extend relative generalized Hamming weights and relative dimension/length profile, respectively, and relative generalized rank weights and relative dimension/intersection profile, respectively.

Relative generalized Hamming weights of cyclic codes

Relative generalized Hamming weights (RGHWs) of a linear code with respect to a linear subcode determine the security of the linear ramp secret sharing scheme based on the linear codes. They can be used to express the information leakage of the secret when some keepers of shares are corrupted. Cyclic codes are an interesting type of linear codes and have wide applications in communication and storage systems. In this paper, we investigate the RGHWs of cyclic codes of two nonzeros with respect to its irreducible cyclic subcodes. We give two formulae for RGHWs of the cyclic codes. As applications of the formulae, explicit examples are computed. Moreover, RGHWs of cyclic codes in the examples are very large, comparing with the generalized Plotkin bound of RGHWs. So it guarantees very high security for the secret sharing scheme based on the dual codes.

Flags of almost affine codes and the two-party wire-tap channel of type II

We describe a two-party wire-tap channel of type II in the framework of almost affine codes. Its cryptological performance is related to some relative profiles of a pair of almost affine codes. These profiles are analogues to relative generalized Hamming weights in the linear case.

Refined analysis of RGHWs of code pairs coming from Garcia-Stichtenoth’s second tower

Asymptotically good sequences of ramp secret sharing schemes were given in [5] by using one-point algebraic geometric codes defined from asymptotically good towers of function fields. Their security is given by the relative generalized Hamming weights of the corresponding codes. In this paper we demonstrate how to obtain refined information on the RGHWs when the codimension of the codes is small. For general codimension, we give an improved estimate for the highest RGHW.

Relative generalized Hamming weights of <i>q</i>-ary Reed-Muller codes

Coset constructions of q-ary Reed-Muller codes can be used to store secrets on a distributed storage system in such a way that only parties with access to a large part of the system can obtain information while still allowing for local error-correction. In this paper we determine the relative generalized Hamming weights of these codes which can be translated into a detailed description of the information leakage [ 2 , 21 , 18 , 9 ]

Some bounds for the relative generalized Hamming weights of some evaluation codes

Abstract In this paper we find some bounds for the relative generalized Hamming weights of some codes parameterized by a set of monomials of the same degree. Also we compare the relative generalized Hamming weights of the codes CX(d) and CX′(d) when X′ is embedded in X. We use these results to obtain some lower bounds for the relative generalized Hamming weights of the codes parameterized by the edges of any connected bipartite graph with bipartition (V1, V2) and where |V1| = n1, |V2| = n2, in terms of the relative generalized Hamming weights of the codes associated to the projective tori Tn₁-1 and Tn₂-1.

Bounds for generalized Hamming weights of general AG codes

We present a good bound for the generalized Hamming weights of multi-point evaluation and differential AG codes. It is a natural generalization of the order bound for one-point AG codes. As an example, we determine the third generalized Hamming weight of one particular two-point evaluation Hermitian code. Then we extend the bound for the relative generalized Hamming weights, which are essential measures of the performance of secret sharing schemes based on linear codes. The access structure of one particular secret sharing scheme based on the previous two-point Hermitian code and its subcode is analyzed using the new bound.

Relative Generalized Hamming Weights of One-Point Algebraic Geometric Codes

Security of linear ramp secret sharing schemes can be characterized by the relative generalized Hamming weights of the involved codes. In this paper we elaborate on the implication of these parameters and we devise a method to estimate their value for general one-point algebraic geometric codes. As it is demonstrated, for Hermitian codes our bound is often tight. Furthermore, for these codes the relative generalized Hamming weights are often much larger than the corresponding generalized Hamming weights.

On the relative generalized hamming weights of a 4-dimensional linear code and a subcode with dimension one

Finite projective geometry method is effectively used to study the relative generalized Hamming weights of 4-dimensional linear codes, which are divided into 9 classes in order to get much more information about the relative generalized Hamming weights, and part of the relative generalized Hamming weights of a 4-dimensional linear code with a 1-dimensional subcode are determined.

New code equivalence based on relative generalized Hamming weights

Code equivalence is a basic concept in coding theory. The well-known theorem by MacWilliams gives a sufficient condition for code equivalence. Recently the MacWilliams theorem has been generalized, by Fan, Liu and Puig, making use of the generalized Hamming weights (GHWs). In this paper, we will present a further generalization of the MacWilliams theorem. Our result extends both the MacWilliams theorem and the result by Fan, Liu and Puig. We will first define “relative subcodes” of a linear code, based on the relative generalized Hamming weights (RGHWs) which is a generalization of the GHWs; and then establish a method based on finite projective geometry to characterize relative subcodes. Using this method, we will prove our main result.

On the Relative Generalized Hamming Weights of Linear Codes and their Subcodes

We first present an equivalent definition of relative generalized Hamming weights of a linear code and its subcodes, and we develop a method using finite projective geometry. Making use of the equivalent definition and the projective-geometry method, all of the relative generalized Hamming weights of a 3-dimensional q-ary linear code and its subcodes will be determined.

The relative generalized Hamming weight of linear q-ary codes and their subcodes

In this article, some properties of the relative generalized Hamming weight (RGHW) of linear codes and their subcodes are developed with techniques in finite projective geometry. The relative generalized Hamming weights of almost all 4-dimensional q-ary linear codes and their subcodes are determined.