MDS array codes for correcting a single criss-cross error

Abstract

Abstract-We present a family of Maximum-Distance Separable (MDS) array codes of size ( 1) ( 1), a prime number, and minimum criss-cross distance 3, i.e., the code is capable of correcting any row or column in error, without a priori knowledge of what type of error occurred. The complexity of the encoding and decoding algorithms is lower than that of known codes with the same error-correcting power, since our algorithms are based on exclusive-OR operations over lines of different slopes, as opposed to algebraic operations over a finite field. We also provide efficient encoding and decoding algorithms for errors and erasures.

Codes With Locality in the Rank and Subspace Metrics

“…Note that it is not possible to correct such an error pattern using a code equipped with the Hamming metric. On the other hand, rank-metric codes are well known for their ability to effectively correct rankerrors [12], [13].…”

“…Errors and erasures that affect a limited number of rows and/or columns are usually referred to as crisscross patterns [12], [13]. (See Fig.…”

List decoding of crisscross error patterns

“…Afterwards, add a third non-zero diagonal to each of the "valid" matrices, i.e., consider vecdiag(e (0) , e (1) , e (2) ), for all e (2) ∈ L H 2 . There are at most H · C (q; η 2 , d, τ ) such matrices.…”

“…A similar statement as Lemma 1 holds for any three non-zero diagonals. Thus, when we discard all matrices 2014 IEEE International Symposium on Information Theory vecdiag(e (0) , e (1) , e (2) ) of cover weight greater than τ , at most C (q; η 3 , d, τ ) matrices remain. We continue this strategy until we reach L H m−1 and we finally obtain a list L C of size |L C | ≤ C containing all matrices E of cover weight at most τ ≤ min{τ H , τ C − 1} such that (R − E) ∈ C(C, m).…”

“…1 illustrates a binary matrix and its two minimum covers. Codes in the cover metric based on maximum distance separable (MDS) constituent codes were constructed by Roth in [11], one-error-correcting codes and their decoding were shown by Blaum and Bruck in [2], and constructions for special parameters were given in [9] by Lund, Gabidulin and Honary and in [14] by Sidorenko. The codes from [11], [9] achieve the Singleton-like upper bound on the dimension, see [5], [11].…”