# 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 X n matrices over F such that every nonzero matrix in C has rank μ. It is first shown that the dimension of such Σarray codes must satisfy the Singleton-like bound k ≤n(n-μ + l). 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 T Ñ”c, given a “received” array T + E, where rank(E)= t ≤(μ-1)/2. Maximum-rank array codes can be used for decoding crisscross errors in n X 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. © 1991 IEEE