In this paper, we establish the necessary and sufficient conditions for quasi-cyclic (QC) codes with index even to be symplectic self-orthogonal. Subsequently, we present the lower and upper bounds on the minimum symplectic distances of a class of 1-generator QC codes and their symplectic dual codes by decomposing code spaces. As an application, we construct many new binary symplectic self-orthogonal QC codes with excellent parameters, leading to 117 record-breaking quantum error-correction codes.

Theq-ary maximum distance separable (MDS) array codes, especiallyq= 2v, have important applications in the storage systems to prevent data loss. At now, almost all algebraic constructions of such codes are carried out over the quotient ring Rm= Fq[x]/⟨xm+1⟩ or its subring Fq[x]/⟨xm−1+xm−2+· · ·+1⟩. In this paper, we consider the construction of MDS array codes derived form Rm,λ= Fq[x]/⟨xm+ λ⟩, where λ ∈ F∗q= Fq\{0}, gcd(m, q) = 1. By giving λ-GCD Constraint of polynomials and the punctured λ-twisted circulant matrices, a generic construction and its extensions of such codes are presented. Under this framework, four new explicit constructions are also provided. Specifically, some low-density MDS array codes are obtained in Explicit Constructions I and I ′ . Explicit Constructions II and III provide long MDS array codes that can be applied in the large-scale storage systems. Explicit Construction IV gives MDS array codes both with the optimal repair bandwidth and the lowest update complexity, whose sub-packetization level is smaller than that of codes in [31]. In addition, by the LU factorization, a decoding method for erasures is obtained. For long codes, the syndromes can be computed fast. When the erased number ρ ≤ 3 (the most common errors in the storage systems), its computational complexity is asymptotically optimal. By an example, the new encoder possesses fewer exclusive ORs per data bit than that of MDS array codes constructed over Rmin [29].

The maximum distance separable (MDS) codes are the optimal codes to achieve Singleton bound, providing maximum error tolerance under a given number of parity nodes. Ye and Barg leveraged permutation matrices and Reed-Solomon type codes to devise 7 explicit constructions for constructing MDS array codes with optimal repair property (as known as MSR codes) or even optimal access property. Drawing inspiration from these explicit constructions, we provide two generic constructions for constructing MSR codes from high-rate MDS codes or MDS array codes. In this paper, we introduce the concepts of s-pairwise MDS codes sets and s-pairwise MDS array codes sets. Two generic constructions (Generic Constructions IandII) for constructing the MSR code using thes-pairwise MDS codes sets or thes-pairwise MDS array codes sets are given. Constructions 1 to 3 proposed by Ye and Barg can be regarded as some special cases ofGeneric Construction I, and Constructions 1 to 3 proposed by Li et al., can be regarded as some special cases ofGeneric Construction II. It is worth mentioning thatGeneric Construction IIcan be applied to any finite field, including the binary field. We also demonstrate how to obtain thes-pairwise MDS code sets and thes-pairwise MDS array code sets from a high-rate MDS code or MDS array code over$\mathbb {F}_{q}$. We obtain a novel class of MSR codes by utilizing the MDS array codes provided by Lv et al., as component codes according toGeneric Construction II. As a byproduct of Constructions 4 to 7, we obtain a new class of MSR codes with optimal access property. Using two types of sets$\Gamma _{1}$and$\Gamma _{2}$with the property that matrices commute, we present several new constructions for MSR codes with the optimal access property over any finite field. In this paper, compared with the constructions over binary field proposed by Li et al., the sub-packetization of our constructions applicable to the binary field is significantly reduced.

It is well-known that erasure codes have been widely applied in the distributed storage systems (DSSs). This paper concentrates on the update bandwidth of erasure codes, which is defined as the system network IO consumption required when a node is updated. Based on the OUB-codes [1] and MDS array codes [2], we introduce the array-OUB-codes which are vertical MDS array codes with triple fault tolerant and minimal update bandwidth. In addition, according to our experiments implemented by C++ program, our array-OUB-codes achieve better encoding and decoding performance compared with OUB-codes. Moreover, through the update bandwidth IO test, our array-OUB-codes can increase the average throughput by 37.5% compared with the RS scheme in [3].

In this short work, a new explicit construction of$q-\mathbf{ary}$MDS array codes with multiple parities will be provided, whose code lengths can be up to$q^{m-1}$, where$m-1$is the size of subpackage. As far as we know, this may be the first explicit construction of practical MDS array codes with such long code lengths for general$q$. In addition, to demonstrate the applicability of our MDS array codes, by the LU factorization of Vandermonde matrices, we present an efficient decoding method aimed at the erased errors, whose computational complexity is$O(m^{2})$in total. Furthermore, if one stores a small number of polynomials in advance or computes the syndrome in a scheduled algorithm, the decoding efficiency of these MDS array codes can be further improved.

EVENODD+ codes are binary maximum distance separable (MDS) array codes for correcting double disk failures in RAID-6 with asymptotically optimal encoding/decoding/update complexities. However, the number of bits stored in each disk of EVENODD+ codes should be an odd number minus one. In this paper, we present a new construction of EVENODD+ codes that have more flexible parameters. The number of bits stored in each disk of our codes is an odd minus one times any positive integer. Moreover, our codes not only have asymptotically optimal encoding/decoding/update complexities but also have lower encoding/decoding/update complexities than the existing EVENODD+ codes.

Cauchy maximum distance separable (MDS) array codes are widely used in storage systems that can support high fault tolerance with systematic form. However, Cauchy MDS array codes suffer from high encoding/decoding complexities and limited parameters. In this paper, we first construct new Cauchy MDS array codes with circulant structure over quotient rings that have flexible sub-packetization. We show that our codes can support more parameters than the existing Cauchy MDS array codes. The sub-packetization of our codes is a multiple of prime minus one, while the sub-packetization of the existing Rabin-Like codes and circulant Cauchy codes is prime minus one. Second, we propose a fast decoding method by designing new division algorithm and employing the LU decomposition of Cauchy matrix for the proposed Cauchy MDS array codes. We show that the decoding complexity of our codes is nearly the same as new Rabin-Like codes [16], and lower than all the other Cauchy MDS array codes.

MDS array codes have been extensively studied due to their applications in storage systems. In this paper, we first propose a novel method of constructing MDS array codes by deleting one row and one column from the circulant matrices associated to some polynomials. Several new classes of MDS array codes with flexible parameters are constructed. In particular, we give a new algebraic presentation of the Blaum-Roth codes with sparser parity-check matrices. We also obtain a family of MDS array codes over finite fields with even characteristics whose parity-check matrices have the lowest density. Furthermore, based on these new MDS array codes, we give a general construction of optimal locally repairable array codes (LRACs) achieving the Singleton-type bound. Additionally, we obtain some new optimal LRACs of long lengths. Finally, we present a scheduled algorithm for syndrome computations of binary optimal LRACs with redundancy 4, which can tolerate three failures. The number of XORs per data bit required in our algorithm approaches 2 as the length approaches infinity, which is the same as the MDS codes tolerating three failures. However, the number of nodes required during the repair of a failed node in our optimal LRACs is only about half of that in MDS array codes.

This paper aims to construct optimal quaternary additive codes with non-integer dimensions. Firstly, we propose combinatorial constructions of quaternary additive constant-weight codes, alongside additive generalized anticode construction. Subsequently, we propose generalized Construction X, which facilitates the construction of non-integer dimensional optimal additive codes from linear codes. Then, we construct ten classes of optimal quaternary non-integer dimensional additive codes through these two methods. As an application, we also determine the optimal additive$[n,3.5,n-t]_{4}$codes for all t with variable n, except for$t=6,7,12$.

In this short paper, we will provide a new explicit construction of binary MDS array codes with triple parities from their parity-check matrices, which contains array codes with array number 8 (8 bits=1 byte). In addition, to demonstrate the applicability of our MDS array codes, we present an effective decoding method aimed at the erased errors. Furthermore, a fast encoding algorithm of our extended MDS array codes is also explored, whose computational complexity is 2 XORs per bit when their code lengths approach infinity.

From the perspective of parity-check matrices, we present new constructions of$q$-ary maximum distance separable (MDS) array codes with multiple parities. Applying these constructions, some new types of MDS array codes with array numbers$m-\tau $can be derived, where${\mathrm{ gcd}}(m,q)=1$. Moreover, an explicit construction of binary MDS array codes is also presented. Compared to the existing MDS array codes, one important characteristic of these codes is that their available code lengths are much longer, which is suitable for large-scale storage systems. In some particular cases, the maximum code lengths of these codes and their extension can be up to$2^{m-\tau }$and$2^{m-\tau }+1$(or$2^{m-\tau }+2$), respectively. Moreover, to demonstrate the applicability of our constructed MDS array codes, we present an effective generic decoding method for the erased errors. In particular, when there are no more than three erasures occurring, a scheduled algorithm for the syndrome computation of our explicit construction is further proposed, whose computational complexity is asymptotically optimal. Furthermore, this algorithm can be directly applied to the encoding procedure of their extended form. The simulation shows that our new MDS array codes have better encoding and decoding performances than the corresponding extended RS codes coupled with different algorithms.

In this paper, we firstly present a new construction of binary maximum distance separable (MDS) array codes, from which some types of new MDS array codes of minimum distance 4 with array dimension (p−1)×(ℓ+2) can be deduced. Based on the construction, binary locally repairable array codes (LRACs) of minimum distance 4 are also explored, whose array dimension is (p−1)×2ℓ and column locality is ℓ − 1. Particularly, when 2 is a primitive root module p, a scheduled algorithm for syndrome computation of the LRACs is proposed, which converges to 2 XORs per data bit when ℓ approaches infinity.