Maximum-rank array codes and their application to crisscross error correction

Abstract

A µ-[n × n, k] array code C over a field F is a k-dimensional linear space of n × n matrices over F such that every nonzero matrix in C has rank ≥ µ. It is first shown that the dimension of such array codes must satisfy the Singleton-like bound k ≤ n(n−µ+1). A family of so-called maximum-rank µ-[n × n, k = n(n − µ + 1)] array codes is then constructed over every finite field F and for every n and µ, 1 ≤ µ ≤ n. A decoding algorithm is presented for retrieving every Γ ∈ C, given a "received" array Γ + E, where rank(E) = t ≤ (µ − 1)/2. Maximum-rank array codes can be used for decoding crisscross errors in n × n bit arrays, where the erroneous bits are confined to a number t of rows or columns (or both). Our construction proves to be optimal also for this model of errors, which can be found in a number of applications, such as memory chip arrays or magnetic tape recording. Finally, it is shown that the behavior of linear spaces of matrices is quite unique compared with the more general case of linear spaces of n × n × • • • × n hyper-arrays.

A Rank-Metric Approach to Error Control in Random Network Coding

“…Proof: By the Singleton bound for subspace codes (13), (14) and the fact that C achieves the Singleton bound for rank-metric codes (8), we have…”

“…A matrix code is also commonly known as an array code when it forms a linear space over F q [13]. The rate of a matrix code…”

“…The rank metric was introduced in coding theory by Delsarte [11]. Codes for the rank metric were largely developed by Gabidulin [12] (see also [13]). An important feature of the coding theory for the rank metric is that it supports many of the powerful concepts and techniques of classical coding theory, such as linear and cyclic codes and corresponding decoding algorithms [12]- [15].…”

Vector network coding based on subspace codes outperforms scalar linear network coding

“…The Singleton-like upper bound for rank-metric codes [6], [21], [38] implies that for any [m×n, k, δ] R q code, we have that k ≤ max{m, n}(min{n, m} − δ + 1). Codes which attain this bound with equality are known for all feasible parameters [6], [21], [38].…”

“…Codes which attain this bound with equality are known for all feasible parameters [6], [21], [38]. They are called maximum rank distance (MRD) codes and denoted by MRD[m × n, δ] q .…”