This paper investigates the use of redundancy and self repairing against node failures indistributed storage systems using a novel non-MDS erasure code. In replication method, accessto one replication node is adequate to reconstruct a lost node, while in MDS erasure codedsystems which are optimal in terms of redundancy-reliability tradeoff, a single node failure isrepaired after recovering the ...

We present some results on almost maximum distance separable (AMDS) codes and Griesmer codes of dimension 4 over F5. We prove that no AMDS code of length 13 and minimum distance 5 exists, and we give a classification of some AMDS codes. Moreover, we classify the projective strongly optimal Griesmer codes over F5 of dimension 4 for some values of the minimum distance.

A group code defined over a group G is a subset of Gn which forms a group under componentwise group operation. The well known matrix characterization of MDS (Maximum Distance Separable) linear codes over finite fields is generalized to MDS group codes over abelian groups, using the notion of quasideterminants defined for matrices over non-commutative rings.

A subset S of {0, 1, . . . , 2t − 1}n is called a t-fold MDS code if every line in each of n base directions contains exactly t elements of S. The adjacency graph of a t-fold MDS code is not connected if and only if the characteristic function of the code is the repetition-free sum of the characteristic functions of t-fold MDS codes of smaller lengths. In the case t = 2, the theory has the foll...

In this paper two-dimensional convolutional codes with finite support are considered, i.e., convolutional codes whose codewords have compact support indexed in N and take values in F, where F is a finite field. The main goal of this work is to analyze the (free) distance properties of this type of codes of rate 1/n and degree δ. We first establish an upper bound on the maximum possible distance...

AbsrructAn ( n, k, d) linear code over F = GF( q) is said to be ntcrximunt dktunce separable (MDS) if d = n k + 1. It is shown that an (II, k, FI k + 1) generalized Reed-Solomon code such that 2 I k 5 n 1 ((I 1)/2] (k + 3 if q is even) can be extended by one digit while preserving the MDS property if and only if the resulting extended code is also a generalized Reed-Solomon code. It follows tha...

The Main Conjecture on maximum distance separable (MDS) codes states that, except for some special cases, the maximum length of a q-ary linear MDS code of is q+1. This conjecture does not hold true for near maximum distance separable codes because of the existence of q-ary near-MDS elliptic codes having length bigger than q+1. An interesting related question is whether a near-MDS elliptic code ...

The union of two disjoint (n, 4n−1, 2) MDS codes in {0, 1, 2, 3}n is a double-MDS-code. If the adjacency graph of the double-MDS-code is not connected, then the double-code can be decomposed into double-MDS-codes of smaller lengths. If the graph has more than two connected components, then the MDS codes are also decomposable. The result has an interpretation in terms of n-quasigroups of order 4...

Abstract—We propose random relaying of MDS codeword blocks in cooperative multi-hop networks and evaluate its performance over random error channels by means of an absorbing Markov chain. The proposed scheme can apply to cooperative multi-hop networks with the arbitrary number of relay nodes at each hop stage and require no feedback channels between nodes. In the proposed scheme, a message bloc...

Maximum-distance separable (MDS) convolutional codes form an optimal family of convolutional codes, the study of which is of great importance. There are very few general algebraic constructions of MDS convolutional codes. In this paper, we construct a large family of unit-memory MDS convolutional codes over Fq with flexible parameters. Compared with previous works, the field size q required to ...