Correlations between the ranks of submatrices and weights of random codes

Date
2009
Authors
Klyachko, A. A.
Özen, I.
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Source Title
Finite Fields and Their Applications
Print ISSN
1071-5797
Electronic ISSN
1090-2465
Publisher
Volume
15
Issue
4
Pages
497 - 516
Language
English
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Abstract

The results of our study are twofold. From the random matrix theory point of view we obtain results on the rank distribution of column submatrices. We give the moments and the covariances between the ranks (q- rank) of such submatrices. We conjecture the counterparts of these results for arbitrary submatrices. The case of higher correlations gets drastically complicated even in the case of three submatrices. We give a formula for the correlation of ranks of three submatrices and a conjecture for its closed form. From the code theoretical point of view our study yields the covariances of the coefficients of the weight enumerator of a random code. Particularly interesting is that the coefficients of the weight enumerator of a code with random parity check matrix are uncorrelated. We give a conjecture for the triple correlations between the coefficients of the weight enumerator of a random code. © 2009 Elsevier Inc. All rights reserved.

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