On the ideal associated to a linear code

This article aims to explore the bridge between the algebraic structure of a linear code and the complete decoding process. To this end, we associate a specific binomial ideal $I_+(\mathcal C)$ to an arbitrary linear code. The binomials involved in the reduced Gr\"obner basis of such an ideal relative to a degree-compatible ordering induce a uniquely defined test-set for the code, and this allows the description of a Hamming metric decoding procedure. Moreover, the binomials involved in the Graver basis of $I_+(\mathcal C)$ provide a universal test-set which turns out to be a set containing the set of codewords of minimal support of the code

Multiple Quantitative Trait Analysis Using Bayesian Networks

Models for genome-wide prediction and association studies usually target a single phenotypic trait. However, in animal and plant genetics it is common to record information on multiple phenotypes for each individual that will be genotyped. Modeling traits individually disregards the fact that they are most likely associated due to pleiotropy and shared biological basis, thus providing only a partial, confounded view of genetic effects and phenotypic interactions. In this paper we use data from a Multiparent Advanced Generation Inter-Cross (MAGIC) winter wheat population to explore Bayesian networks as a convenient and interpretable framework for the simultaneous modeling of multiple quantitative traits. We show that they are equivalent to multivariate genetic best linear unbiased prediction (GBLUP), and that they are competitive with single-trait elastic net and single-trait GBLUP in predictive performance. Finally, we discuss their relationship with other additive-effects models and their advantages in inference and interpretation. MAGIC populations provide an ideal setting for this kind of investigation because the very low population structure and large sample size result in predictive models with good power and limited confounding due to relatedness.Comment: 28 pages, 1 figure, code at http://www.bnlearn.com/research/genetics1

Universal Gr\"obner Bases for Binary Linear Codes

Each linear code can be described by a code ideal given as the sum of a toric ideal and a non-prime ideal. In this way, several concepts from the theory of toric ideals can be translated into the setting of code ideals. It will be shown that after adjusting some of these concepts, the same inclusion relationship between the set of circuits, the universal Gr\"obner basis and the Graver basis holds. Furthermore, in the case of binary linear codes, the universal Gr\"obner basis will consist of all binomials which correspond to codewords that satisfy the Singleton bound and a particular rank condition. This will give rise to a new class of binary linear codes denoted as Singleton codes.Comment: Accepted for publication in IJPA

Graver Bases and Universal Gr\"obner Bases for Linear Codes

Two correspondences have been provided that associate any linear code over a finite field with a binomial ideal. In this paper, algorithms for computing their Graver bases and universal Gr\"obner bases are given. To this end, a connection between these binomial ideals and toric ideals will be established.Comment: 18 page

On Binomial Ideals associated to Linear Codes

Recently, it was shown that a binary linear code can be associated to a binomial ideal given as the sum of a toric ideal and a non-prime ideal. Since then two different generalizations have been provided which coincide for the binary case. In this paper, we establish some connections between the two approaches. In particular, we show that the corresponding code ideals are related by elimination. Finally, a new heuristic decoding method for linear codes over prime fields is discussed using Gr\"obner bases

Space-time coding techniques with bit-interleaved coded modulations for MIMO block-fading channels

The space-time bit-interleaved coded modulation (ST-BICM) is an efficient technique to obtain high diversity and coding gain on a block-fading MIMO channel. Its maximum-likelihood (ML) performance is computed under ideal interleaving conditions, which enables a global optimization taking into account channel coding. Thanks to a diversity upperbound derived from the Singleton bound, an appropriate choice of the time dimension of the space-time coding is possible, which maximizes diversity while minimizing complexity. Based on the analysis, an optimized interleaver and a set of linear precoders, called dispersive nucleo algebraic (DNA) precoders are proposed. The proposed precoders have good performance with respect to the state of the art and exist for any number of transmit antennas and any time dimension. With turbo codes, they exhibit a frame error rate which does not increase with frame length.Comment: Submitted to IEEE Trans. on Information Theory, Submission: January 2006 - First review: June 200