$X$-secure and $T$-private information retrieval (XSTPIR) is a variant of private information retrieval where data security is guaranteed against collusion among up to $X$ servers and the user's retrieval privacy is guaranteed against collusion among up to $T$ servers. Recently, researchers have constructed XSTPIR schemes through the theory of algebraic geometry codes and algebraic curves, with the aim of obtaining XSTPIR schemes that have higher maximum PIR rates for fixed field size and $X,T$ (the number of servers $N$ is not restricted). The mainstream approach is to employ curves of higher genus that have more rational points, evolving from rational curves to elliptic curves to hyperelliptic curves and, most recently, to Hermitian curves. In this paper, we propose a different perspective: with the shared goal of constructing XSTPIR schemes with higher maximum PIR rates, we move beyond the mainstream approach of seeking curves with higher genus and more rational points. Instead, we aim to achieve this goal by enhancing the utilization efficiency of rational points on curves that have already been considered in previous work. By introducing a family of bases for the polynomial space $\text{span}_{\mathbb{F}_q}\{1,x,\dots,x^{k-1}\}$ as an alternative to the Lagrange interpolation basis, we develop two new families of XSTPIR schemes based on rational curves and Hermitian curves, respectively. Parameter comparisons demonstrate that our schemes achieve superior performance. Specifically, our Hermitian-curve-based XSTPIR scheme provides the largest known maximum PIR rates when the field size $q^2\geq 14^2$ and $X+T\geq 4q$. Moreover, for any field size $q^2\geq 28^2$ and $X+T\geq 4$, our two XSTPIR schemes collectively provide the largest known maximum PIR rates.

Maximum distance separable (MDS) codes are widely used in distributed storage, but naively repairing a single failure in an $(n,k)$ MDS code requires downloading the full contents of $k$ surviving nodes. Minimum storage regenerating (MSR) codes, introduced by Dimakis et al., minimize repair bandwidth while preserving the MDS property by contacting $d>k$ helper nodes and downloading only a fraction of each helper. For scalar MDS codes, Guruswami and Wootters established a linear repair framework, and Tamo, Ye, and Barg subsequently gave the first explicit Reed-Solomon (RS) codes achieving the MSR point. Their construction yields RS-MSR codes with subpacketization $\ell=s\prod_{i=1}^n p_i$, where $s=d+1-k$ and the distinct primes $p_i$ satisfy $p_i\equiv 1\pmod{s}$. In this paper, we show that this congruence condition is not intrinsic to the RS repair problem. We develop a basis-transformation approach to the construction of repair-enabling subspaces. The approach consists of three deterministic operations -- Euclidean Square Partition, Transposition, and Column Aggregation -- which construct the required repair-enabling subspaces directly from the standard monomial basis of the repair field. Consequently, we obtain RS-MSR codes with subpacketization $\ell=s\prod_{i=1}^n p_i$ for arbitrary distinct primes $p_i>s$. For fixed $s$, this improves the subpacketization of the Tamo--Ye--Barg construction by a factor asymptotic to $φ(s)^{n+\mathrm{o}(n)}$, where $φ(\cdot)$ denotes Euler's totient function.

Levenshtein first introduced the sequence reconstruction problem in $2001$. In the realm of combinatorics, the sequence reconstruction problem is equivalent to determining the value of $N(n,d,t)$, which represents the maximum size of the intersection of two metric balls of radius $t$, given that the distance between their centers is at least $d$ and the sequence length is $n$. In this paper, We present a lower bound on $N(n,3,t)$ for $n\geq \max\{13,t+8\}$ and $t \geq 4$. For $t=4$, we prove that this lower bound is tight. This settles an open question posed by Pham, Goyal, and Kiah, confirming that $N(n,3,4)=20n-166$ for all $n \geq 13$.

Error-correcting codes are essential for ensuring fault tolerance in modern distributed data storage systems. However, in practice, factors such as the failure rates of storage devices can vary significantly over time, resulting in changes to the optimal code parameters. To reduce storage cost while maintaining efficiency, Maturana and Rashmi introduced a theoretical framework known as code conversion, which enables dynamic adjustment of code parameters according to device performance. In this paper, we focus exclusively on the bounds and constructions of generalized merge-convertible codes. First, we establish a new lower bound on the access cost when the final code is an (r, δ)-LRC. This bound unifies and generalizes all previously known bounds for merge conversion, where the initial and final codes are either LRCs or MDS codes. We then construct a family of access-optimal MDS convertible codes by leveraging subgroups of the automorphism group of a rational function field. It is worth noting that our construction is also per-symbol read access-optimal. Next, We extend our MDS-based construction to design access-optimal convertible codes that enable conversion between (r, δ)-LRCs with parameter regimes that have not been previously reported in the literature. Finally, using the parity-check matrix approach, we present a construction of access-optimal convertible codes that enable merge conversion from MDS codes to an (r, δ)-LRC. To the best of our knowledge, this is the first explicit optimal construction of code conversion between MDS codes and LRCs. All of our constructions are performed over finite fields whose sizes grow linearly with the code length.

Binary cyclic codes are worth studying due to their applications and theoretical importance. It is an important problem to construct an infinite family of cyclic codes with large minimum distancedand dual distanced⊥. In recent years, much research has been devoted to improving the lower bound ond, some of which have exceeded the square-root bound. The constructions presented recently seem to indicate that when the minimum distance increases, the minimum distance of its dual code decreases. In this paper, we focus on the new constructions of binary cyclic codes with lengthn=2m− 1, dimension nearn/2 and both relatively large minimum distance and dual distance. Whenmis even, we construct a family of binary cyclic codes with parameters [2m− 1, 2m−1± 1, d], whered≥ 2m/2− 1 andd⊥≥ 2m/2. Both the minimum distance and the dual distance are significantly better than the previous results. Whenmis the product of two distinct primes, we construct some cyclic codes with dimensionsk= (n+1)/2 andd>n/log2n, where the lower bound on the minimum distance is much larger than the square-root bound. Whenmis odd, we present two families of binary [2m−1, 2m−1,d] cyclic codes withd≥ 2(m+1)/2−1,d⊥≥2(m+1)/2andd≥ 2(m+3)/2− 15,d⊥≥ 2(m−1)/2respectively, which leads thatd·d⊥can reach 2nasymptotically. To the best of our knowledge, except for the punctured binary Reed-Muller codes, there is no other construction of binary cyclic codes that reaches this bound.

Locally repairable codes (LRCs) have attracted considerable attention in recent years due to their practical applications in distributed storage systems and their inherent theoretical significance. In this paper, we focus on alphabet-dependent optimal LRCs. We first establish some new bounds for the code length, dimension and minimum distance of$(r, \delta)$-LRCs, respectively. In particular, we propose an explicit bound, the Griesmer-like bound, which extends the classical Griesmer bound to accommodate locality constraints. This is also the generalization of a Griesmer-like bound proposed by Hao et al. in 2020. Furthermore, by lengthening the$q$-ary first-order Reed-Muller code, we present two classes of (2,$q$) -LRCs that achieve this newly derived Griesmer-like bound. It is worth noting that the first class of$(2, q)$-LRCs does not meet the classical Griesmer bound as linear codes. To the best of our knowledge, it is the first class of Griesmer-like LRCs that are not Griesmer codes. This also implies that our Griesmer-like bound for$(r, \delta)$-LRCs is non-trivial.

In this paper, we focus on Singleton-optimal$(r,\delta)$LRCs with disjoint local repair groups. We provide an improved bound for the length of q-ary Singleton-optimal$(r,\delta)$LRCs based on the parity-check matrix approach. Specifically, for$d \geq 3\delta $, we prove that$n\le O(q^{\delta })$when$d-3\delta \lt r\le d-2\delta +1$. We also show that the code length$n\le q+\delta +2$when$r=2$and$d=3\delta +2$. We present a sufficient and necessary condition for the existence of Singleton-optimal$(n,k,d;r,\delta)$LRCs with disjoint local repair groups, where the minimum distance satisfies$3\delta +1\le d \le 3\delta +2$and locality$r=2$. This condition imposes an upper bound on the code length,$n\le O(q^{2})$, and indicates the existence of a code length approximately given by$n\approx \sqrt {2}q$when$d=3\delta +1$and$r=2$. Finally, we utilize blocking sets to provide a general construction of Singleton-optimal$(n,k,d=2\delta +2,r=2,\delta)$LRC with code length$n\approx O\left ({{q^{\frac {h+1}{h}}}}\right)$for any$h\ge 3$. To the best of our knowledge, this is the first family of Singleton-optimal$(n,k,d=2\delta +2,r=2,\delta)$LRC with super-linear code length.

Transformers have recently emerged as effective neural decoders, capable of capturing complex interdependencies and delivering strong performance. Cyclic codes, including BCH codes, Reed-Solomon codes, and Reed-Muller codes, are fundamental in practical channel coding applications. In this work, we propose a novel approach for Transformer-based neural decoders specifically designed for cyclic codes. By leveraging the relational properties among the parity-check matrix, Tanner graph, and mask matrix, we integrate these with the inherent algebraic structure of cyclic codes. We then identify a cyclic shift reuse property that can be effectively applied to both parameter matrices and self-attention matrices. Extensive simulations on BCH codes demonstrate that our method reduces the total number of parameters by up to 64.4% compared to prior Transformer-based decoders while achieving comparable decoding performance. Additionally, our method decreases the computational cost of self-attention by up to 10.6%. Finally, through experiments across various dimensions, we explore the effect of embedding dimensions on performance and provide insights into the relationship between embedding dimensions and code positions.

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].

Suppose a distributed storage system containingMfiles is replicated acrossNservers, and a user wants to privately retrieve one file by accessing the servers such that the identity of the retrieved file is kept secret from any subset of up toTservers, where each file can be viewed as a vector over theq-ary finite field Fq. A scheme designed for this purpose is called aT-private information retrieval (T-PIR) scheme. We consider the problem of explicitly constructing capacity-achievingT-PIR schemes over small finite fields. In this paper, we first provide a general framework for constructing explicit capacity-achievingT-PIR schemes for all parameters, which only relies on an MDS array matrix with a special information set. To construct such an MDS array matrix, we propose a new family of matrices over finite fields, called the generalized minor matrices of the Moore matrix, and establish a series of key identities. By combining favourable properties of generalized minor matrices with our framework, we construct an explicit capacityachievingT-PIR scheme with optimal sub-packetization over the field Fq, as small as possible, for three classes of parametersN, T,M≥ 3. Specifically, the first class of construction works for allN=d(2t− 1),T=dt, and the field sizeqis the least prime power satisfyingqt−1≥N. Moreover, this construction generalizes the scheme proposed by Xu and Wang in 2022, which only considers the case ofN=d(2t−1),T= dt with 2t−1≥N. For allN=d(2t+ 1),T= dt, our secondT-PIR scheme is the first explicit construction, and the field sizeqis the least prime power satisfyingqt≥N, which is the smallest field size among all known explicit capacity-achievingT-PIR schemes. Particularly, when 2t≥N, the field size of such constructions can be reduced to 2. In the case ofN= 4sandT= 2s+ 1, our scheme is the first one to reduce the field size toq= 2. Compared with all known explicitly capacity-achievingT-PIR schemes, the required field of our schemes has the smallest size.

The deep holes of a linear code are the vectors achieving maximum error distance to the code. There has been a lot of work on the deep holes of Reed -Solomon codes. In this paper, we consider the deep holes of a class of twisted Reed -Solomon codes. The covering radius and a standard class of deep holes of twisted Reed-Solomon codes TRS$k(\mathcal{A},\ \eta)$are obtained for a general evaluation set$\mathcal{A}\subseteq \mathbb{F}_{q}$• Furthermore, when$q=2^{m}\geq 8$, we prove that there are no other deep holes of the full-length twisted Reed-Solomon codes TRS$k(\mathbb{F}_{q},\ \eta)$for$\displaystyle \frac{3}{4}q-1\leq k\leq{q}-4$, and we also completely determine their deep holes for$q-3\leq k\leq q-1$•

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.

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.

An$(n, k, d; r)_{q}$-locally repairable code (LRC) is called a Singleton-optimal LRC if it achieves the Singleton-type bound. Analogous to the classical MDS conjecture, the maximal length problem of Singleton-optimal LRCs has attracted a lot of attention in recent years. In this paper, we give an improved upper bound for the length of q-ary Singleton-optimal LRCs with disjoint repair groups such that$(r+1)\mid n$based on the parity-check matrix approach. In particular, for any Singleton-optimal$(n, k, d; r)_{q}$-LRCs, we show that: 1)$n\le q+d-4$, when$r=2$and$d=3e+8$with$e\ge 0$; 2)$n\leq (r+1)\left \lfloor {{\frac {2(q^{2}+q+1)}{r(r+1)} +e+1}}\right \rfloor $, when$d\ge 8$and$\max \left \{{{3,\frac {d-e-6}{e+1}}}\right \}\le r\le \frac {d-e-3}{e+1}$for any$0\le e\le \left \lfloor {{\frac {d-6}{4} }}\right \rfloor $. Furthermore, we establish equivalent connections between the existence of Singleton-optimal$(n,k,d;r)_{q}$-LRCs for$d=6, r=3$and$d=7, r=2$with disjoint repair groups and some subsets of lines in finite projective space with certain properties. Consequently, we prove that the length of q-ary Singleton-optimal LRCs with minimum distance$d=6$and locality$r=3$is upper bounded by$O(q^{1.5})$. We construct Singleton-optimal$(8\le n\le q+1,k,d=6,r=3)_{q}$-LRC with disjoint repair groups such that$4\mid n$and determine the exact value of the maximum code length for some specific q. We also prove the existence of$(n, k, d=7; r=2)_{q}$-Singleton-optimal LRCs for$n \approx \sqrt {2}q$.

Cyclic codes are an important class of linear codes. Bounding the minimum distance of cyclic codes is a long-standing research topic in coding theory, and several well-known and basic results have been developed on this topic. Recently, locally repairable codes (LRCs) have attracted much attention due to their repair efficiency in large-scale distributed storage systems. In this paper, by employing the singleton procedure technique, we first provide a sufficient condition for bounding the minimum distance of cyclic codes with typical defining sets. Secondly, by considering a specific case, we establish a connection between bounds for the minimum distance of cyclic codes and solutions to a system of inequalities. This connection leads to the derivation of new bounds, including some with general patterns. In particular, we provide three new bounds with general patterns, one of which serves as a generalization of the Betti-Sala bound. Finally, we present a generalized lower bound for a special case and construct several families of (2, δ)-LRCs with unbounded length and minimum distance 2δ. It turns out that these LRCs are distance-optimal, and their parameters are new. To the best of our knowledge, this work represents the first construction of distance-optimal (r, δ)-LRCs with unbounded length and minimum distance exceedingr+ δ - 1.

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.

Locally repairable codes (LRCs) have attracted a lot of attentions due to their applications in distributed storage systems. In this paper, we provide new constructions of optimal (2, δ)-LRCs. Firstly, by the techniques of finite geometry, we present a sufficient condition to guarantee a punctured simplex code to be a (2, δ)-LRC. Secondly, by using characteristic sums over finite fields and Krawtchouk polynomials, we construct several families of LRCs with new parameters. All of our new LRCs are optimal with respect to the generalized Cadambe-Mazumdar bound.

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.

There has been a lot of works about constructing optimal LRCs via cyclic codes because of their elegant algebraic structure and efficient encoding procedure. Constructing optimal cyclic LRCs with large code lengths for relatively large minimum distances has been an attractive problem. Recently, Fang et al. firstly constructed two classes of q-ary cyclic Singleton-optimal 2-LRCs with length n > q + 1 and minimum distance d = 6 in [23]. In this paper, we generalize the constructions to the (r,δ)-LRCs. Specifically, we obtain two classes of optimal cyclic (2,δ)-LRCs with length $n = \frac{{(\delta + 1)(q + 1)}}{{{2^t}}}$ and minimum distance 2δ + 2, two classes of almost optimal cyclic (2,δ)-LRCs with length $n = \frac{{(\delta + 1)(q + 1)}}{{{2^t}}}$ and minimum distance 2δ +1, where t is a non-negative integer.

Although federated learning improves privacy of training data by exchanging local gradients or parameters rather than raw data, the adversary still can leverage local gradients and parameters to obtain local training data by launching reconstruction and membership inference attacks. To defend against such privacy attacks, many noises perturbed methods (like differential privacy or CountSketch matrix) have been widely designed. However, the strong defence ability and high learning accuracy of these schemes cannot be ensured at the same time, which will impede the wide application of FL in practice (especially for medical or financial institutions that require both high accuracy and strong privacy guarantee). To overcome this issue, we propose an efficient model perturbation method for federated learning to defend against reconstruction and membership inference attacks launched by curious clients. On the one hand, similar to the differential privacy, our method also selects random numbers as perturbed noises added to the global model parameters, and thus it is very efficient and easy to be integrated in practice. Meanwhile, the random selected noises are positive real numbers and the corresponding value can be arbitrarily large, and thus the strong defence ability can be ensured. On the other hand, unlike differential privacy or other perturbation methods that cannot eliminate added noises, our method allows the server to recover the true aggregated gradients by eliminating the added noises. Therefore, our method does not hinder learning accuracy at all. Extensive experiments demonstrate that for both regression and classification tasks, our method achieves the same accuracy as non-private approaches and outperforms the state-of-the-art defence schemes. Besides, the defence ability of our method against reconstruction and membership inference attack is significantly better than the state-of-the-art related defence schemes.

Locally repairable codes (LRCs) have emerged as an important coding scheme in distributed storage systems (DSSs) with relatively low repair cost by accessing fewer non-failure nodes. Theoretical bounds and optimal constructions of LRCs have been widely investigated. Optimal LRCs via cyclic codes provide significant benefit of elegant algebraic structure and efficient encoding procedure. In this paper, we continue to consider the constructions of optimal LRCs via cyclic codes with longer code length. Specifically, we first obtain two classes of Singleton-optimal cyclic LRCs with length$n=3(q+1)$when$3\vert (q-1)$and$q$is even, and length$n=\frac{3}{2}(q+1)$when$3\vert (q-1)$and$q$is odd, respectively. To the best of our knowledge, this is the first construction of q-ary cyclic Singleton-optimal LRCs with length$n > q+1$and minimum distance$d\geq 5$. By using cyclic codes as well, we construct a new family of perfect LRCs with$d=5$, which generalize the result of Goparaju and Calderbank.

Repair locality has been an important metric in a distributed storage system (DSS). Erasure codes with small locality are more popular in a DSS, which means fewer available nodes participating in the repair process of failed nodes. Locally repairable codes (LRCs) as a new coding scheme have given more rise to the system performance and attracted a lot of interest in the theoretical research in coding theory. The particular concern among the research problems is the bounds and optimal constructions of LRCs. The problem of optimal constructions of LRCs includes the most important case of Singleton-optimal LRCs whose minimum distance achieves the Singleton-like bound, which is the core consideration in this paper. In this work, we first of all derive an improved and general upper bound on the code length of Singleton-optimal LRCs with minimum distance d = 5, 6, some known constructions are shown to exactly achieve our new bound, which verifies its tightness. For locality r = 2 and distance d = 6, we construct three newSingleton-optimal LRCs whose code length n = 3(q + 1), n = 3(q + √q + 1) and n = 3(2q - 4), respectively. Moreover, we obtain a complete characterization for Singletonoptimal LRCs with r = 2 and d = 6. Such characterization has established an important connection between the existence of Singleton-optimal LRCs and that of a special subset of lines of finite projective plane P G(2, q), thus provides a methodology for constructing LRCs with longer length based on any advance on finite projective plane P G(2, q). In the end, we employ the well-known line-point incidence matrix and Johnson bounds for constant weight codes to derive tighter upper bounds on the code length. These new bounds further help us to prove that some of the previous Singleton-optimal constructions or their extensions achieve the longest possible code length for q = 3, 4, 5, 7. It's worth noting that all of our Singleton-optimal constructions possess small locality r = 2, which are attractive in a DSS.

The parameters of a q-ary MDS Euclidean self-dual codes are completely determined by its length and the construction of MDS Euclidean self-dual codes with new length has been widely investigated in recent years. In this paper, we give a further study on the construction of MDS Euclidean self-dual codes via generalized Reed-Solomon (GRS) codes and their extended codes. The main idea of our construction is to choose suitable evaluation points such that the corresponding (extended) GRS codes are Euclidean self-dual. Firstly, we consider the evaluation set consists of two disjoint subsets, one of which is based on the trace function, the other one is a union of a subspace and its cosets. Then four new families of MDS Euclidean self-dual codes are constructed. Secondly, we give a simple but useful lemma to ensure that the symmetric difference of two intersecting subsets of finite fields can be taken as the desired evaluation set. Based on this lemma, we generalize our first construction and provide two new families of MDS Euclidean self-dual codes. Finally, by using two multiplicative subgroups and their cosets which have nonempty intersection, we present three generic constructions of MDS Euclidean self-dual codes with flexible parameters. Several new families of MDS Euclidean self-dual codes are explicitly constructed.

Locally repairable codes (LRCs) with locality r were introduced to recover an erased code symbol by accessing at most r other code symbols. An LRC achieving the well-known Singleton-type bound is called an optimal LRC. Constructing optimal LRCs has been a hot topic of coding theory in recent years. Similar to the famous MDS conjecture, the maximum code length of an optimal LRC has been investigated by Guruswami et al. (TIT2019) and some constructions of optimal LRCs with large code length are also presented by Jin (TIT2019) and Xing and Yuan (arXiv2018). In this paper, we consider the maximum code length of optimal LRCs with minimum distance 6 and locality 2. Firstly, we give a complete characterization for optimal LRCs with d = 6 and r = 2, which shows that the existence of such an LRC is equivalent to the existence of a special subset of lines of finite projective plane PG(2, q). Based on this characterization, we generalize the results of Chen et al. (ISIT2018) and obtain two new constructions of optimal (n, k, d = 6; r = 2)-LRCs with n = 3(q + √q + 1) and n = 3(2q -4), respectively. By using the techniques of line-point incidence matrix and Johnson bound, we show that the code length of any q-ary optimal LRCs with d = 6 and r = 2 must be bounded by O(q1.5). To the best of our knowledge, both of the code length of our new constructions and upper bounds are better than previously known ones. Moreover, we also determine the exact value of the maximum code length of q-ary optimal LRCs with d = 6 and r = 2 for q = 4, 5.

Linear codes with locality, called locally repairable codes (LRCs), have been applied in distributed storage systems (DSSs) to minimize the number of storage nodes to be downloaded during repairing a failed node. A linear code has locality r if one can recover an erased code symbol by accessing at most r other code symbols. Bounds and constructions of LRCs have been widely investigated in recent years. In this paper, we first propose the definition of perfect LRCs, whose dimension k achieves the Hamming-type bound proposed by Wang et al. (TIT2019). Then we establish important connections of the existence of LRCs with finite geometry and finite fields, and two systematic constructions of perfect LRCs are obtained. Rewriting the Hamming-type bound by the property of integers, we present a new construction of k-optimal LRCs achieving this bound, which have longer code length than the previously known ones.

In this paper, we construct several classes of maximum distance separable (MDS) codes via generalized Reed-Solomon (GRS) codes and extended GRS codes, where we can determine the dimensions of their Euclidean hulls or Hermitian hulls. It turns out that the dimensions of Euclidean hulls or Hermitian hulls of the codes in our constructions can take all or almost all possible values. As a consequence, we can apply our results to entanglement-assisted quantum error-correcting codes (EAQECCs) and obtain several new families of MDS EAQECCs with flexible parameters. The required number of maximally entangled states of these MDS EAQECCs can take all or almost all possible values. Moreover, several new classes of q-ary MDS EAQECCs of length n > q+1 are also obtained.

Locally repairable codes (LRCs) are introduced in distributed storage systems due to their low repair overhead. An LRC is called optimal if its minimum distance attains the Singleton-like upper bound. Chen et al. (2018) recently studied the constructions of optimal (r, δ)-LRCs with length n | (q+1) and (r + δ - 1) | n, where many classes of optimal cyclic constructions were obtained. In this paper, by employing constacyclic MDS codes, we construct seven classes of optimal (r, δ)-LRCs with new parameters. After adding these new optimal LRCs via constacyclic codes, we have completely obtained all optimal (r, δ)-LRCs with length n | (q + 1) and (r + δ - 1) | n for all possible parameters for the completeness in the coding theory. It is worth noting that the optimal constacyclic LRCs with new parameters provide more alternatives to cyclic LRCs in the practical demands of distributed storage systems, where specific values of n, k, r, and δ are required. Moreover, constacyclic LRCs also possess the encoding and decoding efficiency as cyclic LRCs.

In this paper, we consider the problem for which lengths a maximum distance separable (MDS) Euclidean self-dual code over Fq exists. This problem is completely solved for the case where q is even. For q is odd, some q-ary MDS Euclidean self-dual codes were obtained in the literature. In this paper, we construct six new classes of q-ary MDS Euclidean self-dual codes by using generalized Reed-Solomon (GRS for short) codes and extended GRS codes.

It is an important task to construct quantum maximum-distance-separable (MDS) codes with good parameters. In the present paper, we provide six new classes of$q$-ary quantum MDS codes by using generalized Reed–Solomon (GRS) codes and Hermitian construction. The minimum distances of our quantum MDS codes can be larger than$\frac {q}{2}+1$. Three of these six classes of quantum MDS codes have longer lengths than the ones constructed in[1]and[2], hence some of their results can be easily derived from ours via the propagation rule. Moreover, some known quantum MDS codes of specific lengths can be seen as special cases of ours and the minimum distances of some known quantum MDS codes are also improved as well.

Prakash et al. [2] introduced the concept of (r, δ) locally repairable codes ((r, δ)-LRCs for short) for tolerating multiple failed nodes. An (r, δ)-LRC is called optimal if it achieves the Singleton-type bound. In this paper, inspired by the work of [3], we firstly construct two classes of optimal cyclic (r, δ)-LRCs with unbounded lengths (i.e., lengths of these codes are independent of the alphabet size) and minimum distances δ+1 or δ + 2, which generalize the results about the δ = 2 case given in [3]. Secondly, with a slightly stronger condition, we present a construction of optimal cyclic (r, δ)-LRCs with unbounded length and larger minimum distance 2δ. Furthermore, when δ = 3, we provide another class of optimal cyclic (r, 3)-LRCs with unbounded length and larger minimum distance 6.

Optimal (r, δ) locally repairable codes ((r, δ)-LRCs for short) with length n | (q+1) have been studied in [5] and [6]. In this paper, along with their ideas, by using cyclic or constacyclic codes, we construct three classes of such LRCs with new parameters which are not obtained in [5] and [6]. Thus, optimal (r, δ)-LRCs with length n | (q+1) and (r + δ - 1) | n are completely determined for all possible parameters.