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Binary quasi-reversible BCH codes whose defining set contains consecutive elements from negative to positive integers have received considerable attention in recent years due to their efficient decoding suitable for a wide range of applications. This paper combines the concept of the weight and quasi-reversible structures to introduce two subclasses of BCH codes: odd-like/even-like quasi-reversible BCH codes. The step-by-step decoding of these codes is developed as follows: First, the weight evaluation of a received polynomial is able to judge whether the number of errors is odd or even, which helps to simplify the decoding processes. Second, based on Chiò’s pivotal condensation process which can be easily implemented in a parallel computing architecture, the determinant calculation of the band matrix instead of Peterson’s matrix in column-echelon form is faster. Third, a newly proposed non-monic error-locator polynomial is sparser than the conventional ones. As a consequence, the theoretical analysis and experimental results validate potential benefits in requiring fewer finite field additions and multiplications used in the decoding of binary odd-like/even-like quasi-reversible BCH codes up to half the minimum distance when compared with the narrow-sense BCH codes with small error-correcting capability. [ABSTRACT FROM AUTHOR]

The recently introduced polar codes constitute a breakthrough in coding theory due to their capacity-achieving property. This goes hand in hand with a quasilinear construction, encoding, and successive cancellation list decoding procedures based on the Plotkin construction. The decoding algorithm can be applied with slight modifications to Reed-Muller or eBCH codes, that both achieve the capacity of erasure channels, although the list size needed for good performance grows too fast to make the decoding practical even for moderate block lengths. The key ingredient for proving the capacity-achieving property of Reed-Muller and eBCH codes is their group of symmetries. It can be plugged into the concept of Plotkin decomposition to design various permutation decoding algorithms. Although such techniques allow to outperform the straightforward polar-like decoding, the complexity stays impractical. In this paper, we show that although invariance under a large automorphism group is valuable in a theoretical sense, it also ensures that the list size needed for good performance grows exponentially. We further establish the bounds that arise if we sacrifice some of the symmetries. Although the theoretical analysis of the list decoding algorithm remains an open problem, our result provides an insight into the factors that impact the decoding complexity. [ABSTRACT FROM AUTHOR]

Locally recoverable codes play a crucial role in distributed storage systems. Many studies have only focused on the constructions of optimal locally recoverable codes with regard to the Singleton bound. The aim of this paper is to construct optimal binary locally recoverable codes meeting the alphabet-dependent bound. Using a general framework for linear codes associated to a set, we provide a new approach to constructing binary locally recoverable codes with locality 2. We turn the problem of designing optimal binary locally recoverable codes into constructing a suitable set. Several constructions of optimal binary locally recoverable codes are proposed by this new method. Finally, we propose constructions of optimal binary locally recoverable codes with locality 2 and locality parameters $(\text {r},\delta)$ by Griesmer codes. [ABSTRACT FROM AUTHOR]

Recently, some high-rate maximum distance separable (MDS) array codes were designed to optimally repair a single failed node by connecting all the surviving nodes. However, in practical systems, sometimes not all the surviving nodes are available. To facilitate the practical storage system, a few constructions of $(n,k)$ MDS array codes with the property that any single failed node can be optimally repaired by accessing any $d$ surviving nodes (i.e., minimum-storage regenerating (MSR) codes) have been proposed, where $d\in [k+1:n-1)$. However, all high-rate MDS array codes with this property either have large sub-packetization levels or are not explicit for all the parameters. To address these issues, we propose a generic transformation that can convert any $(n',k')$ MDS array/scalar code to another $(n=n'-\delta,k=k'-\delta)$ MDS array code with the optimal repair property and optimal access property for an arbitrary set of two nodes, while the repair efficiency of the remaining $n-2$ nodes can be kept, where $2\le \delta \le n'-k'$. By recursively applying the generic transformation to an MDS scalar code multiple times, we get a high-rate MDS array code with the optimal repair property and the optimal access property for all nodes, which outperforms previous known high-rate MDS array codes in terms of either the sub-packetization level or the flexibility of the parameters. [ABSTRACT FROM AUTHOR]