MDS codes have garnered significant attention due to their wide applications in practice. To date, most known MDS codes are equivalent to Reed-Solomon codes. The construction of non-Reed-Solomon (non-RS) type MDS codes has emerged as an intriguing and important problem in both coding theory and finite geometry. Although some constructions of non-RS type MDS codes have been presented in the literature, the parameters of these MDS codes remain subject to strict constraints. In this paper, we introduce a general framework of constructing [n,k] MDS codes using the idea of selecting a suitable set of evaluation polynomials and a set of evaluation points such that all nonzero polynomials have at mostk–1 zeros in the evaluation set. Moreover, these MDS codes can be proved to be non-Reed-Solomon by computing their Schur squares. Furthermore, several explicit constructions of non-RS MDS codes are given by converting to combinatorial problems. As a result, new families of non-RS MDS codes with much more flexible lengths can be obtained and most of them are not covered by the known results.

MDS codes play a central role in practice due to their broad applications. To date, most known MDS codes are generalized Reed–Solomon (GRS) codes, leaving codes that are not equivalent to GRS codes comparatively less understood. Studying this non-GRS regime is therefore of intrinsic theoretical interest, and is also practically relevant since the strong algebraic structure of GRS codes can be undesirable in cryptographic settings. Among the known non-GRS codes, twisted generalized Reed–Solomon (TGRS) codes and Roth–Lempel codes are two representative families of non-GRS codes that have attracted significant attention. Though substantial work has been devoted to the construction and structural analysis of TGRS and Roth–Lempel codes, comparatively little attention has been paid to their decoding, and many problems remain open. In this paper, we propose list and unique decoding algorithms for TGRS codes and Roth–Lempel codes based on the Guruswami–Sudan algorithm. Under suitable parameter conditions, our algorithms achieve near-linear running time in the code length, improving upon the previously best-known quadratic-time complexity. Our TGRS decoder supports fixed-rate TGRS codes with up toO(n2)twists, substantially extending prior work that only handled the single-twist case. For Roth–Lempel codes, we provide what appears to be the first efficient decoder. Moreover, our list decoders surpass the classical unique-decoding radius for a broad range of parameters. Finally, we incorporate algebraic manipulation detection (AMD) codes into the list-decoding framework, enabling recovery of the correct message from the output list with high probability.

In the realm of modern digital communication, cryptography, and signal processing, binary sequences with good correlation properties play a pivotal role. In the literature, considerable efforts have been dedicated to constructing good binary sequences of various lengths. As a consequence, numerous constructions of good binary sequences have been put forward. However, the majority of known constructions leverage the multiplicative cyclic group structure of finite fields Fpn, wherepis a prime andnis a positive integer. Recently, the authors made use of the cyclic group structure of all rational places of the rational function field over the finite field Fpn, and firstly constructed good binary sequences of lengthpn+ 1 via cyclotomic function fields over Fpnfor any primep[8], [10]. This approach has paved a new way for constructing good binary sequences. Motivated by the above constructions, we exploit the cyclic group structure of rational points of elliptic curves to design a family of binary sequences of length 2n+1+twith low correlation for many given integers |t| ⩽ 2(n+2)/2. Specifically, for any positive integerdwith gcd(d; 2n+1+t) = 1, we introduce a novel family of binary sequences of length 2n+1+t, sizeqd−1− 1, correlation bounded by (2d+ 1) · 2(n+2)/2+ |t|, and large linear complexity via elliptic curves.

Sequences with a low correlation have very important applications in communications, cryptography, and compressed sensing. In the literature, many efforts have been made to construct good sequences with various lengths, where binary sequences attract great attention. As a result, various constructions of good binary sequences have been proposed. However, most of the known constructions made use of the multiplicative cyclic group structure of finite field$\mathbb {F}_{p^{n}}$for a prime$p$and a positive integer$n$. In fact, all$p^{n}+1$rational places including the place at infinity of the rational function field over$\mathbb {F}_{p^{n}}$can form a cyclic structure under an automorphism of order$p^{n}+1$. In this paper, we make use of this cyclic structure to provide an explicit construction of binary sequences with a low correlation of length$p^{n}+1$via cyclotomic function fields over$\mathbb {F}_{p^{n}}$for any odd prime$p$. Each family of binary sequences has size$p^{n}-2$and its correlation is upper bounded by$4+\lfloor 2\cdot p^{n/2}\rfloor $. To the best of our knowledge, this is the first construction of binary sequences with a low correlation of length$p^{n}+1$for odd prime$p$. Moreover, our sequences can be constructed explicitly and have competitive parameters.

A pure quantum state of$n$parties associated with the Hilbert space$\mathbb {C}^{d_{1}}\otimes \mathbb {C} ^{d_{2}}\otimes \cdots \otimes \mathbb {C} ^{d_{n}}$is called$k$-uniform if all the reductions to$k$-parties are maximally mixed. The$n$partite system is called homogenous if the local dimensions$d_{1}=d_{2}=\cdots =d_{n}$, while it is called heterogeneous if the local dimensions are not all equal.$k$-uniform sates play an important role in quantum information theory. There are much progress in characterizing and constructing$k$-uniform states in homogeneous systems. However, the study of entanglement for heterogeneous systems is much more challenging than that for the homogeneous case. There are very few results known for the$k$-uniform states in heterogeneous systems for$k>3$. We present two general methods to construct$k$-uniform states in the heterogeneous systems for general$k$. The first construction is derived from the error correcting codes by establishing a connection between irredundant mixed orthogonal arrays and error correcting codes. We can produce many new$k$-uniform states such that the local dimension of each subsystem can be a prime power. The second construction is derived from a matrix$H$meeting the condition that$H_{A\times \bar {A}}+H^{T}_{\bar {A}\times A}$has full rank for any row index set$A$of size$k$. These matrix construction can provide more flexible choices for the local dimensions, i.e., the local dimensions can be any integer (not necessarily prime power) subject to some constraints. Our constructions imply that for any positive integer$k$, one can construct$k$-uniform states of a heterogeneous system in many different Hilbert spaces.

Locally Repairable codes (LRCs) have gained significant interest due to applications in distributed storage systems since they enable systems to recover a failed node by accessing few other active nodes. In particular, LRCs with large availability are highly desirable for parallel reading of hot data. In addition to the above applications in distributed storage systems, it was shown by Fazeli et al. that an LRC with large availability can produce a good private information retrieval (PIR) code which allows to reduce the storage overhead of a PIR protocol. Roughly speaking, one can obtain a good PIR code as long as there exists an LRC with large availability. One of the main tasks in studying PIR codes is to design a$t$-server PIR code with small length for the given dimension. In particular, the construction of binary PIR codes is of great interest. In this paper, we consider a construction of binary LRCs from polynomial evaluations. As a result, a new class of binary LRCs with large availability are obtained. Applying such LRCs to PIR codes, we obtain a new class of binary PIR codes. On one hand, the binary LRCs constructed are new in the sense that the parameter regime is not covered by the known LRCs. On the other hand, the parameters of PIR codes derived from our LRCs outperform the known results in certain parameter regimes and achieve the lower bound given by Fazeli et al. up to an absolute constant.

Due to wide applications of binary sequences with a low correlation to communications, various constructions of such sequences have been proposed in the literature. Many efforts have been made to construct good binary sequences with various lengths. However, most of the known constructions make use of the multiplicative cyclic group structure of finite field$\mathbb {F}_{2^{n}}$for a positive integer$n$. It is often overlooked in this community that all$2^{n}+1$rational places (including “the place at infinity”) of the rational function field over$\mathbb {F}_{2^{n}}$form a cyclic structure under an automorphism of order$2^{n}+1$. In this paper, we make use of this cyclic structure to provide an explicit construction of binary sequences with a low correlation of length$2^{n}+1$via cyclotomic function fields over$\mathbb {F}_{2^{n}}$. Each family of our sequences has size$2^{n}-1$and its correlation is upper bounded by$\lfloor 2^{(n+2)/2}\rfloor $. To the best of our knowledge, this is the first construction of binary sequences with a low correlation of length$2^{n}+1$. Moreover, our sequences can be constructed explicitly and have competitive parameters. In particular, compared with the Gold sequences of length$2^{n}-1$for even$n$, our sequences have a smaller correlation and a larger length although the family size of our sequences is slightly smaller.

In literature, there are several methods to construct Gold sequences. One of the constructions is via the trace function from extension field of \mathbb F2. Estimation of correlation of this construction is based on number of rational points of elliptic curves. In this article, we generalize this construction from finite fields of even characteristic to odd characteristics by using multiplicative quadratic character. Again, estimation of correlation of this construction is based on number of rational points of elliptic curves. Thus, we obtain binary sequences which have more flexibility on length while still possessing low correlation property. Moreover, some of the sequences are optimally balanced.

It is well known that constructing codes with good parameters is one of the most important and fundamental problems in coding theory. Though a great many of good codes have been produced, most of them are defined over alphabets of sizes equal to prime powers. In this article, we provide a new explicit construction of$(q+1)$-ary nonlinear codes via rational function fields, where$q$is a prime power. Our codes are constructed by evaluations of rational functions at all rational places (including the place of “infinity”) of the rational function field. Compared to the rational algebraic geometry codes, the main difference is that we allow rational functions to be evaluated at pole places. After evaluating rational functions from a union of Riemann-Roch spaces, we obtain a family of nonlinear codes with length$q+1$over the alphabet$\mathbb {F}_{q}\cup \{\infty \}$. As a result, our codes have reasonable parameters as they are rather close to the Singleton bound. Furthermore, our codes have better parameters than those obtained from MDS codes via code alphabet restriction or extension. Amazingly, an efficient decoding algorithm can be provided for our codes.

Local Reconstruction Codes (LRCs) allow for recovery from a small number of erasures in a local manner based on just a few other codeword symbols. They have emerged as the codes of choice for large scale distributed storage systems due to the very efficient repair of failed storage nodes in the typical scenario of a single or few nodes failing, while also offering fault tolerance against worst-case scenarios with more erasures. A maximally recoverable (MR) LRC offers the best possible blend of such local and global fault tolerance, guaranteeing recovery from all erasure patterns which are information-theoretically correctable given the presence of local recovery groups. MR LRCs have received much attention recently, with many explicit constructions covering different regimes of parameters. Unfortunately, all known constructions require a large field size. In this work, we develop an approach based on function fields to construct MR LRCs. Our method recovers, and in most parameter regimes improves, the field size of previous approaches. The improvements are modest, but more importantly are obtained in a unified manner via a promising new idea.

Locally repairable codes with more than one recovering set are demanded in the application to distributed storage. For each failure node (or disk), it is desired to have as many recovering sets as possible. In this paper, we make use of automorphisms of rational function fields to construct locally repairable codes with multiple recovering sets. Although we focus on two recovering sets, our construction can be easily generalized to the case of multiple recovering sets. In particular, we obtain a class of locally repairable codes with minimum distance only 1 less than the upper bound.

Locally repairable codes, or locally recoverable codes (LRC for short), are designed for applications in distributed and cloud storage systems. Similar to classical block codes, there is an important bound called the Singleton-type bound for locally repairable codes. In this paper, an optimal locally repairable code refers to a block code achieving this Singleton-type bound. Like classical MDS codes, optimal locally repairable codes carry some very nice combinatorial structures. Since the introduction of the Singleton-type bound for locally repairable codes, people have put tremendous effort into construction of optimal locally repairable codes. There are a few constructions of optimal locally repairable codes in the literature. Most of these constructions are realized via either combinatorial or algebraic structures. In this paper, we apply automorphism group of the rational function field to construct optimal locally repairable codes by considering the group action on projective lines over finite fields. Due to various subgroups of the projective general linear group, we are able to construct optimal locally repairable codes with flexible locality as well as smaller alphabet size comparable to the code length. In particular, we produce new families of q-ary locally repairable codes, including codes of length q+1 via cyclic groups.

Local Reconstruction Codes (LRCs) allow for recovery from a small number of erasures in a local manner based on just a few other codeword symbols. A maximally recoverable (MR) LRC offers the best possible blend of such local and global fault tolerance, guaranteeing recovery from all erasure patterns which are information-theoretically correctable given the presence of local recovery groups. In an $(n,r,h,a)$-LRC, the $n$ codeword symbols are partitioned into $r$ disjoint groups each of which include $a$ local parity checks capable of locally correcting $a$ erasures. MR LRCs have received much attention recently, with many explicit constructions covering different regimes of parameters. Unfortunately, all known constructions require a large field size that exponential in $h$ or $a$, and it is of interest to obtain MR LRCs of minimal possible field size. In this work, we develop an approach based on function fields to construct MR LRCs. Our method recovers, and in most parameter regimes improves, the field size of previous approaches. For instance, for the case of small $r \ll ε\log n$ and large $h \ge Ω(n^{1-ε})$, we improve the field size from roughly $n^h$ to $n^{εh}$. For the case of $a=1$ (one local parity check), we improve the field size quadratically from $r^{h(h+1)}$ to $r^{h \lfloor (h+1)/2 \rfloor}$ for some range of $r$. The improvements are modest, but more importantly are obtained in a unified manner via a promising new idea.

In a paper by Guruswami et al., it was shown that the length n of a q-ary linear locally recoverable code with distance d ≥ 5 is upper bounded by O(dq3). Thus, it is a challenging problem to construct q-ary locally recoverable codes with distance d ≥ 5 and length approaching the upper bound. The same paper also gave an algorithmic construction of q-ary locally recoverable codes with locality r and length n = Ωr(q2) for d = 5 and 6, where Ωr means that the implicit constant depends on locality r. In this paper, we present an explicit construction of q-ary locally recoverable codes of distance d = 5 and 6 via binary constant weight codes. It turns out that 1) our construction is simpler and more explicit and 2) the length of our codes is greater than previously known.

In recent years, self-dual MDS codes have attracted a lot of attention due to theoretical interest and practical importance. Similar to self-dual MDS codes, self-dual near MDS (NMDS for short) codes have nice structures as well. From both theoretical and practical points of view, it is natural to study self-dual NMDS codes. Although there has been lots of work on NMDS codes in literature, little is known for self-dual NMDS codes. It seems more challenging to construct self-dual NMDS codes than self-dual MDS codes. The only work on construction of self-dual NMDS codes shows existence of q-ary self-dual NMDS codes of length q - 1 for odd prime power q or length up to 16 for some small primes q with q ≤ 197. In this paper, we make use of properties of elliptic curves to construct selfdual NMDS codes. It turns out that, as long as 2|q and n is even with 4 ≤ n ≤ q + 12√qJ - 2, one can construct a self-dual NMDS code of length n over Fq. Furthermore, for odd prime power q, there exists a self-dual NMDS code of length n over Fq if q ≥ 4n+3x (n + 3)2.

In 1964, Massey introduced a class of codes with complementary duals which are called linear complimentary dual (LCD) codes. He showed that LCD codes have applications in communication system, side-channel attack and so on. LCD codes have been extensively studied in literature. On the other hand, MDS codes form an optimal family of classical codes which have wide applications in both theory and practice. The main purpose of this paper is to give an explicit construction of several classes of LCD MDS codes, using tools from algebraic function fields. We exemplify this construction and obtain several classes of explicit LCD MDS codes for the odd characteristic case.

Minimum storage regenerating codes have minimum storage of data in each node and therefore are maximal distance separable (for short) codes. Thus, the number of nodes is upper-bounded by 2b, where ú is the bits of data stored in each node. From both theoretical and practical points of view (see the details in Section 1), it is natural to consider regenerating codes that nearly have minimum storage of data, and meanwhile, the number of nodes is unbounded. One of the candidates for such regenerating codes is an algebraic geometry code. In this paper, we generalize the repairing algorithm of Reed-Solomon codes given by Guruswami and Wotters to algebraic geometry codes and present a repairing algorithm for arbitrary one-point algebraic geometry codes. By applying our repairing algorithm to the one-point algebraic geometry codes based on the Garcia- Stichtenoth tower, one can repair a code of rate 1 - e and length n over Fqwith bandwidth (n - 1)(1 - τ) log q for any e = 2(τ-1/2)logq with a real τ ∈ (0, 1/2). In addition, storage in each node for an algebraic geometry code is close to the minimum storage. Due to nice structures of Hermitian curves, repairing of Hermitian codes is also investigated. As a result, we are able to show that algebraic geometry codes are regenerating codes with good parameters.

It was shown by Massey that linear complementary dual (LCD) codes are asymptotically good. In 2004, Sendrier proved that LCD codes meet the asymptotic Gilbert-Varshamov (GV) bound. Until now, the GV bound still remains to be the best asymptotical lower bound for LCD codes. In this paper, we show that an algebraic geometry code over a finite field of even characteristic is equivalent to an LCD code and consequently there exists a family of LCD codes that are equivalent to algebraic geometry codes and exceed the asymptotical GV bound.

A pure quantum state is called k-uniform if all its reductions to k-qudit are maximally mixed. We investigate the general constructions of k-uniform pure quantum states of n subsystems with d levels. We provide one construction via symmetric matrices and the second one through the classical error-correcting codes. There are three main results arising from our constructions. First, we show that for any given even n ≥ 2, there always exists an n/2-uniform n-qudit quantum state of level p for sufficiently large prime p. Second, both constructions show that there exist k-uniform n-qudit pure quantum states such that k is proportional to n, i.e., k = Ω(n) although the construction from symmetric matrices in general outperforms the one by error-correcting codes. Third, our symmetric matrix construction provides a positive answer to the open question on whether there exists a 3-uniform n-qudit pure quantum state for all n ≥ 8. In fact, we can further prove that, for every k, there exists a constant Mksuch that there exists a k-uniform n-qudit quantum state for all n ≥ Mk. In addition, by using the concatenation of algebraic geometry codes, we give an explicit construction of k-uniform quantum state when k tends to infinity.

The Reed-Muller (RM) code, encoding n-variate degree-d polynomials over Fqfor dqn, has a relative distance 1 - d/q and can be list decoded from a 1- O(√d/q) fraction of errors. In this paper, for d ≪ q, we give a length-efficient puncturing of such codes, which (almost) retains the distance and list decodability properties of the RM code, but has a much better rate. Specifically, when q = Ω(d2/ε2), we give an explicit rate Ω (ε/d!) puncturing of RM codes, which have a relative distance at least (1 - √ε) and efficient list decoding up to (1 - √ε) error fraction. This almost matches the performance of random puncturings, which work with the weaker field size requirement q = Ω(d/ε2). We can also improve the field size requirement to the optimal (up to constant factors) q = Ω(d/ε), at the expense of a worse list decoding radius of 1-ε1/3and rate Ω (ε/d!). The first of the above tradeoffs is obtained by substituting for the variables functions with carefully chosen pole orders from an algebraic function field; this leads to a puncturing for which the RM code is a subcode of a certain algebraic-geometric code (which is known to be efficiently list decodable). The second tradeoff is obtained by concatenating this construction with a Reed-Solomon-based multiplication friendly pair, and using the list recovery property of algebraic-geometric codes.

A linear complementary dual (LCD) code is a linear code with complimentary dual. LCD codes have been extensively studied in literature. On the other hand, maximum distance separable (MDS) codes are an important class of linear codes that have found wide applications in both theory and practice. However, little is known about MDS codes with complimentary duals. The main purpose of this paper is to construct several classes of MDS codes with complimentary duals, i.e., LCD MDS codes, through generalized Reed-Solomon codes.

Both Maximum Distance Separable and Euclidean self-dual codes have theoretical and practical importance and the study of MDS self-dual codes has attracted lots of attention in recent years. In particular, determining the existence of q-ary MDS self-dual codes for various lengths has been investigated extensively. The problem is completely solved for the case where q is even. This paper focuses on the case where q is odd. We construct a few classes of new MDS self-dual codes through generalized Reed-Solomon codes. More precisely, we show that for any given even length n, we have a q-ary MDS code as long as q ≡ 1 mod 4 and q is sufficiently large (say q ≥ 4n× n2). Furthermore, we prove that there exists a q-ary MDS self-dual code of length n if q = r2and n satisfies one of the three conditions: 1) n ≤ r and n is even; 2) q is odd and n - 1 is an odd divisor of q - 1; and 3) r ≡ 3 mod 4 and n=2tr for any t ≤ (r - 1)/2.

Due to recent applications to communications over powerlines, multilevel flash memories, and block ciphers, permutation codes have received a lot of attention from both coding and mathematical communities. One of the benchmarks for good permutation codes is the Gilbert-Varshamov bound. Although there have been several constructions of permutation codes, the Gilbert-Varshamov bound still remains to be the best asymptotical lower bound except for a recent improvement in the case of constant minimum distance. In this paper, we present an algebraic construction of permutation codes from rational function fields, and it turns out that, for a prime number n of a symbol length, this class of permutation codes improves the Gilbert-Varshamov bound by a factor n asymptotically for a minimum distance d with d = O(√n). Furthermore, for a constant minimum distance d, we improve the Gilbert-Varshamov bound by a factor n as well as the recent one given by Gao et al. by a factor n/log n asymptotically for all sufficiently large n.

Since discovery of Goppa geometric codes, people have been asking the question: are there different constructions of block codes from algebraic curves that give the same parameters as Goppa geometric codes. Despite of great effort by researchers, no such constructions have been found so far. Although in literature, there are many constructions of block code from algebraic curves, most of them are quite different from the one by Goppa in nature and thus they have different parameters. Some of these constructions have the same parameters as Goppa geometric codes, however it was proved that these are essentially the same codes defined by Goppa. In this paper, we solve this question for the case where the characteristic of the ground field is 2, namely, we present a different construction of block codes from algebraic curves that give the same parameters as Goppa geometric codes for the characteristic 2 case.

In this paper, we study the decoding of dual-containing codes from Hermitian tower and applications to quantum codes. The contribution of this paper is threefold. First, we construct the quantum stabilizer codes from the Hermitian tower. Second, we provide a deterministic decoding algorithm with decoding radius that almost achieves the optimal decoding radius, i.e., (1-R)/4 , where R is the rate. Last and most importantly, we present a Monte Carlo algorithm with decoding radius roughly equal to (1-R)/3 , which is beyond the optimal decoding radius (1-R)/4 . There are several features in this paper. First of all, we employ a differential for the Hermitian tower. This differential plays a crucial role for decoding. We also extend our decoding by passing to the constant field extension. This constant field extension makes the decoding work perfectly.

In this paper, we present an algebraic construction of binary codes through rational function fields. We make use of certain multiplicative group of rational functions for our construction. In particular, the point at infinity can be employed in our construction to get codes of length up to q+1, where q is the ground field size. As a result, several new binary constant-weight codes are found and many new binary nonlinear codes are presented.

Guruswami et al. showed that the list-decodability of random linear codes is as good as that of general random codes. In this paper, we further strengthen the result by showing that the list-decodability of random Euclidean self-orthogonal codes is as good as that of general random codes as well, i.e., achieves the classical Gilbert-Varshamov bound. In particular, we show that, for any fixed finite field Fq, error fraction δ ∈ (0,1 - 1/q) satisfying 1 - Hq(δ) ≤ 1/2, and small ε > 0, with high probability a random Euclidean self-orthogonal code over Fqof rate 1 - Hq(δ) - ε is (δ, O(1/ε))-list-decodable. This generalizes the result of linear codes to Euclidean self-orthogonal codes. In addition, we extend the result to list decoding symplectic dual-containing codes by showing that the list-decodability of random symplectic dual-containing codes achieves the quantum Gilbert-Varshamov bound as well. This implies that list-decodability of quantum stabilizer codes can achieve the quantum Gilbert-Varshamov bound. The counting argument on self-orthogonal codes is an important ingredient to prove our result.

Erasure list decoding was introduced to correct a larger number of erasures by outputting a list of possible candidates. In this paper, we consider both random linear codes and algebraic geometry codes for list decoding from erasures. The contributions of this paper are twofold. First, for arbitrary 00 (R and ϵ are independent), we show that with high probability a q-ary random linear code of rate R is an erasure list-decodable code with constant list size qO(1/ϵ)that can correct a fraction 1 - R - ϵ of erasures, i.e., a random linear code achieves the information-theoretic optimal tradeoff between information rate and fraction of erasures. Second, we show that algebraic geometry codes are good erasure list-decodable codes. Precisely speaking, a q-ary algebraic geometry code of rate R from the Garcia-Stichtenoth tower can correct 1 - R - (1/√q - 1) + (1/q) - ϵ fraction of erasures with list size O(1/ϵ). This improves the Johnson bound for erasures applied to algebraic geometry codes. Furthermore, list decoding of these algebraic geometry codes can be implemented in polynomial time. Note that the code alphabet size q in this paper is constant and independent of ϵ.

A curve attaining the Hasse-Weil bound is called a maximal curve. Usually, classical error-correcting codes obtained from a maximal curve have good parameters. However, the quantum stabilizer codes obtained from such classical error-correcting codes via Euclidean or Hermitian self-orthogonality do not always possess good parameters. In this paper, the Hermitian self-orthogonality of algebraic geometry codes obtained from two maximal curves is investigated. It turns out that the stabilizer quantum codes produced from such Hermitian self-orthogonal classical codes have good parameters.

It has been a great challenge to construct new quantum maximum-distance-separable (MDS) codes. In particular, it is very hard to construct the quantum MDS codes with relatively large minimum distance. So far, except for some sparse lengths, all known q-ary quantum MDS codes have minimum distance ≤q/2 + 1. In this paper, we provide a construction of the quantum MDS codes with minimum distance >q/2 + 1. In particular, we show the existence of the q-ary quantum MDS codes with length n = q2+ 1 and minimum distance d for any d q + 1 (this result extends those given in the works of Guardia (2011), Jin et al. (2010), and Kai an Zhu (2012)); and with length (q2+ 2)/3 and minimum distance d for any d (2q+2)/3 if 3|(q + 1). Our method is through Hermitian selforthogonal codes. The main idea of constructing the Hermitian self-orthogonal codes is based on the solvability in Fqof a system of homogenous equations over Fq2.

There have been various constructions of classical codes from polynomial valuations in literature [2], [7], [8], [10], [11]. In this paper, we present a construction of classical codes based on polynomial construction again. One of the features of this construction is that not only the classical codes arisen from the construction have good parameters, but also quantum codes with reasonably good parameters can be produced from these classical codes. In particular, some new quantum codes are constructed (see Examples V.5 and V.6).

In the present paper, we generalize the ideas of code constructions from our previous papers . It turns out that the codes in the previous papers can be viewed as special cases of those in this paper. Moreover, our constructions produce some good codes in terms of their parameters. In particular, some best-known codes can be obtained through our methods. Furthermore, our constructions are explicit and the codes can be easily implemented as shown in the tables of Appendix. Besides, one new code, i.e., a 4-ary [64,15,31]-linear code, is found through our constructions.

In the present paper, we show that if the dimension of an arbitrary algebraic geometry code over a finite field of even characteristic is slightly less than n/2-g with n being the length of the code and g being the genus of the base curve, then it is equivalent to an Euclidean self-orthogonal code. Previously, such results required a strong condition on the existence of a certain differential. We also show a similar result on Hermitian self-orthogonal algebraic geometry codes. As a consequence, we can apply our result to quantum codes and obtain some good quantum codes. In particular, we obtain a q-ary quantum [[q+1,1]]-MDS code for an even power q which is essential for quantum secret sharing.

It is well known that quantum codes can be constructed through classical symplectic self-orthogonal codes. In this paper, we give a kind of Gilbert-Varshamov bound for symplectic self-orthogonal codes first and then obtain the Gilbert-Varshamov bound for quantum codes. The idea of obtaining the Gilbert-Varshamov bound for symplectic self-orthogonal codes follows from counting arguments.

In this paper, we first construct several classes of classical Hermitian self-orthogonal maximum distance separable (MDS) codes. Through these classical codes, we are able to obtain various quantum MDS codes. It turns out that many of our quantum codes are new in the sense that the parameters of our quantum codes cannot be obtained from all previous constructions.