Error and erasure correcting algorithms for rank codes

Abstract

In this paper, transmitted signals are considered as square matrices of the Maximum rank distance (MRD) (n, k, d)-codes. A new composed decoding algorithm is proposed to correct simultaneously rank errors and rank erasures. If the rank of errors and erasures is not greater than the Singleton bound, then the algorithm gives always the correct decision. If it is not a case, then the algorithm gives still the correct solution in many cases but some times the unique solution may not exist.

List and unique error-erasure decoding of interleaved Gabidulin codes with interpolation techniques

“…As a comparison, the average list size calculated with Lemma 6 is I < 1+6.104·10 −5 , the upper bound from Theorem 3 (and therefore the upper bound from (7), [17]) gives P rk(Q) < sk ≤ P rk(R R ) < n − 1 ≤ 0.04632, and the bound from Lemma 9 gives…”

“…In comparison to classical erasure decoders in Hamming metric, we distinguish two types of erasures in rank metric: row erasures and column erasures. This section provides a generalization of our approach to interpolation-based error-erasure decoding of interleaved Gabidulin codes over F q m with n = m. We consider the most general form of row and column erasures as in [7,32] and show how the additional information can be incorporated into our decoding algorithm from the previous sections. Notice that in this section, we consider only n = m. On the one hand, this simplifies the notations, but on the other hand, Lemma 10 only holds for n = m.…”

Convolutional Codes in Rank Metric With Application to Random Network Coding

“…The rank-metric block bounded minimum distance (BMD) error-erasure decoding algorithms from [7], [27] can recon-…”

“…. Table I denotes some Gabidulin codes, which are defined by submatrices of G, their minimum rank distances and their block rank-metric error-erasure BMD decoders (realized e.g., by the decoders from [7], [27]). These BMD decoders decode correctly if (3) is fulfilled for the corresponding minimum rank distance.…”

“…the received matrix after the multi-shot transmission over the operator channel as in (27). The channel transfer matrices A (i) and B (i) are time-variant.…”