On the similarities between generalized rank and Hamming weights and their applications to network coding

Rank weights and generalized rank weights have been proven to characterize error and erasure correction, and information leakage in linear network coding, in the same way as Hamming weights and generalized Hamming weights describe classical error and erasure correction, and information leakage in wire-tap channels of type II and code-based secret sharing. Although many similarities between both cases have been established and proven in the literature, many other known results in the Hamming case, such as bounds or characterizations of weight-preserving maps, have not been translated to the rank case yet, or in some cases have been proven after developing a different machinery. The aim of this paper is to further relate both weights and generalized weights, show that the results and proofs in both cases are usually essentially the same, and see the significance of these similarities in network coding. Some of the new results in the rank case also have new consequences in the Hamming case

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

A General Upper Bound on the Size of Constant-Weight Conflict-Avoiding Codes

Conflict-avoiding codes are used in the multiple-access collision channel without feedback. The number of codewords in a conflict-avoiding code is the number of potential users that can be supported in the system. In this paper, a new upper bound on the size of conflict-avoiding codes is proved. This upper bound is general in the sense that it is applicable to all code lengths and all Hamming weights. Several existing constructions for conflict-avoiding codes, which are known to be optimal for Hamming weights equal to four and five, are shown to be optimal for all Hamming weights in general.Comment: 10 pages, 1 figur

Generalized Hamming weights of affine cartesian codes

In this article, we give the answer to the following question: Given a field $\mathbb{F}$, finite subsets $A_1,\dots,A_m$ of $\mathbb{F}$, and $r$ linearly independent polynomials $f_1,\dots,f_r \in \mathbb{F}[x_1,\dots,x_m]$ of total degree at most $d$. What is the maximal number of common zeros $f_1,\dots,f_r$ can have in $A_1 \times \cdots \times A_m$? For $\mathbb{F}=\mathbb{F}_q$, the finite field with $q$ elements, answering this question is equivalent to determining the generalized Hamming weights of the so-called affine Cartesian codes. Seen in this light, our work is a generalization of the work of Heijnen--Pellikaan for Reed--Muller codes to the significantly larger class of affine Cartesian codes.Comment: 12 Page

Balanced Sparsest Generator Matrices for MDS Codes

We show that given $n$ and $k$, for $q$ sufficiently large, there always exists an $[n, k]_q$ MDS code that has a generator matrix $G$ satisfying the following two conditions: (C1) Sparsest: each row of $G$ has Hamming weight $n - k + 1$; (C2) Balanced: Hamming weights of the columns of $G$ differ from each other by at most one.Comment: 5 page