The capacity of a finite field matrix channel

2022/10/25 by Simon R. Blackburn⋆, Blackburn, Simon R., Jessica Claridge +1
Biochemistry, Genetics and Molecular Biology · Computer Science · #94A40 #Coding theory and cryptography #Combinatorics (math.CO) #Cooperative Communication and Network Coding #DNA and Biological Computing #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT)

paper · pdf · doi:10.48550/arxiv.2210.14100

openalex publication_date 2022/10/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The Additive-Multiplicative Matrix Channel (AMMC) was introduced by Silva, Kschischang and Kötter in 2010 to model data transmission using random linear network coding. The input and output of the channel are n× m matrices over a finite field \mathbbF_q. On input the matrix X, the channel outputs Y=A(X+W) where A is a uniformly chosen n× n invertible matrix over \mathbbF_q and where W is a uniformly chosen n× m matrix over \mathbbF_q of rank t. Silva et al considered the case when 2n≤ m. They determined the asymptotic capacity of the AMMC when t, n and m are fixed and q→∞. They also determined the leading term of the capacity when q is fixed, and t, n and m grow linearly. We generalise these results, showing that the condition 2n≥ m can be removed. (Our formula for the capacity falls into two cases, one of which generalises the 2n≥ m case.) We also improve the error term in the case when q is fixed.

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