Generalized Reed-Solomon Research Articles

Quantum MDS codes with new length and large minimum distance

According to the well-known CSS construction, constructing quantum MDS codes are extensively investigated via Hermitian self-orthogonal generalized Reed-Solomon (GRS) codes. In this paper, given two Hermitian self-orthogonal GRS codes GRSk1(A,vA) and GRSk2(B,vB), we propose a sufficient condition to ensure that GRSk(A∪B,vA∪B) is still a Hermitian self-orthogonal code. Consequently, we first present a new general construction of infinitely families of quantum MDS codes from known ones. Moreover, applying the trace function and norm function over finite fields, we give another two new constructions of quantum MDS codes with flexible parameters. It turns out that the forms of the lengths of our quantum MDS codes are quite different from previous known results in the literature. Meanwhile, the minimum distances of all the q-ary quantum MDS codes are bigger than q/2+1.

Several classes of Galois self-orthogonal MDS codes and related applications

Let q=ph be a prime power and e be an integer with 0≤e≤h−1. e-Galois self-orthogonal codes are generalizations of Euclidean self-orthogonal codes (e=0) and Hermitian self-orthogonal codes (e=h2 and h is even). In this paper, we propose two general methods to construct e-Galois self-orthogonal (extended) generalized Reed-Solomon (GRS) codes. As a consequence, eight new classes of e-Galois self-orthogonal (extended) GRS codes with odd q and 2e|h are obtained. Based on the Galois dual of a code, we also study its punctured and shortened codes. As applications, new e′-Galois self-orthogonal maximum distance separable (MDS) codes for all possible e′ satisfying 0≤e′≤h−1, new e-Galois self-orthogonal MDS codes via the shortened codes, and new MDS codes with prescribed dimensional e-Galois hull via the punctured codes are derived. Moreover, some new q-ary quantum MDS codes with length greater than q+1 and minimum distance greater than q2+1 are obtained.

New MDS self-dual codes from GRS codes and extended GRS codes

Maximum-distance-separable (MDS) codes and self-dual codes are important families of classical codes in coding theory. It is of interest to investigate MDS self-dual codes. The existence of MDS self-dual codes over finite field Fq has been completely solved for q is even. In this paper, for finite field with odd characteristic, we obtain some new results on the existence of MDS self-dual codes by generalized Reed-Solomon (GRS) codes and extended GRS codes.

MDS, Near-MDS or 2-MDS Self-Dual Codes via Twisted Generalized Reed-Solomon Codes

Twisted generalized Reed-Solomon (TGRS) codes are a family of codes that contains a large number of maximum distance separable (MDS) codes that are non-equivalent to generalized Reed-Solomon (GRS) codes. In this paper, we characterize a sufficient and necessary condition that a twisted Reed-Solomon (TRS) code with two twists is MDS; give a sufficient and necessary condition that a TGRS code with two twists is self-dual, and present some constructions of self-dual TGRS codes. These self-dual codes are MDS, NMDS or 2-MDS. Furthermore, we study the non-GRS properties of TGRS codes with two twists and prove that these codes are non-GRS in most cases.

Optimized Distributed Generalized Reed-Solomon Coding with Space-Time Block Coded Spatial Modulation.

We present a well-known generalized Reed–Solomon (GRS) code incorporated with space-time block coded spatial modulation (STBC-SM) for wireless networks, which is capable of enjoying coded cooperation between the source and the relay. In the proposed distributed GRS-coded STBC-SM (DGRSC-STBC-SM) scheme, the source and relay nodes use distinct GRS codes. At the relay, we employ the concept of information selection to choose the message symbols from the source for further encoding. Thus, the codewords jointly constructed by the source and relay are generated at the destination. For achieving the best codeword set at the destination, we propose an optimal algorithm at the relay to select partial symbols from the source. To reduce the computational complexity, we propose a more practical algorithm with low complexity. Monte Carlo simulation results show that the proposed scheme using the low-complexity algorithm can achieve near-optimal error performance. Furthermore, our proposed scheme provides better error performance than its corresponding coded non-cooperative counterpart and the existing Reed–Solomon coded cooperative SM (RSCC-SM) scheme under identical conditions.

Building Capacity-Achieving T-PIR Schemes for Some Parameters Over Binary Field via Subfield Sub-Codes

The T-Private Information Retrieval (T-PIR) problem is that a user wishes to retrieve a single record from N servers, without revealing the identity of the desired record even if the number of colluding servers up to T. Here every server stores all M records and each record can be viewed as a vector over the q-ary finite field Fq. To reduce the field size, a capacity-achieving T-PIR scheme based on some MDS array codes for all possible N > T,M ≥ 2 was constructed by Xu and Zhang in 2019. A key idea in that scheme is to construct some MDS array codes in query phase, whose generator matrices satisfy the recovery property (see (C1) in Example 1, page 6), i.e., there exist some specific columns in each generator matrix such that the number of them is equal to the rank of such matrix. We call the set of each column index of those columns in each generator matrix as the column index set. However, that scheme didn’t give an explicit selection of the column index set satisfying the recovery property. In this paper, let N = d(2t - 1), N > T = dt > 1,M ≥ 3 and d ≥ 1, under the constraint of 2t-1 ≥ N, we present a novel method to construct capacity-achieving T-PIR schemes over the binary field with an explicit selection of the column index set. Moreover, our scheme has sub-packetization N(2t-1)M-2, which is optimal for all linear capacity-achieving T-PIR schemes determined in [38]. The key to our method is that all locators and column multipliers are precisely selected such that a Generalized Reed-Solomon (GRS) code defined by them has a zero-dimensional subfield sub-code. Compared with all the known capacity-achieving T-PIR schemes for the same nontrivial parameters, ours is the first explicit capacity-achieving T-PIR scheme over the binary field.

New Galois Hulls of GRS Codes and Application to EAQECCs

Galois hulls of linear codes have important applications in quantum coding theory. In this paper, we construct some new classes of (extended) generalized Reed-Solomon (GRS) codes with Galois hulls of arbitrary dimensions. We also propose a general method on constructing GRS codes with Galois hulls of arbitrary dimensions from special Euclidean orthogonal GRS codes. Finally, we construct several new families of entanglement-assisted quantum error-correcting codes (EAQECCs) and MDS EAQECCs by utilizing the above results.

Application of Hermitian self-orthogonal GRS codes to some quantum MDS codes

The construction of quantum maximum distance separable (abbreviated to MDS) error-correcting codes has become one of the major concerns in quantum coding theory. In this paper, we further generalize the approach developed in the previous paper, and construct several new classes of Hermitian self-orthogonal generalized Reed-Solomon (GRS) codes. By employing these classical MDS codes, we obtain several classes of quantum MDS codes with large minimum distance. It turns out that our quantum MDS codes exhibited here have less constraints on the selection of code length and some of them have not been constructed before, and in some cases, have larger minimum distance than previous literature. Meanwhile, about half of the distance parameters of our codes are greater than q2+1.

Construction of MDS Euclidean Self-Dual Codes via Two Subsets

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.

Hulls of Generalized Reed-Solomon Codes via Goppa Codes and Their Applications to Quantum Codes

A Goppa code over \Bbb F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q^m</sub> is a well-known subclass of algebraic error-correcting code. If m=1, then it is a generalized Reed-Solomon(GRS) code and its dual code is called a GRS code via a Goppa code. In this paper, we give a necessary and sufficient condition that the dual codes of GRS codes via (expurgated) Goppa codes are also GRS codes via Goppa codes. Under the above condition, we show that the hulls of GRS codes via Goppa codes are still GRS codes via Goppa codes. As an application, we characterize LCD GRS codes and self-dual GRS codes under the above condition. Some numerical examples are also presented to illustrate our main results. Moreover, we also apply our result to entanglement-assisted quantum error correcting codes (EAQECCs) and obtain two new families of MDS EAQECCs with arbitrary parameters.

Construction of long MDS self-dual codes from short codes

In this paper, we propose a mechanism on how to construct long MDS self-dual codes from short ones. These codes are special types of generalized Reed-Solomon (GRS) codes or extended generalized Reed-Solomon codes. The main tool is utilizing additive structure or multiplicative structure on finite fields. By applying this method, more MDS self-dual codes can be constructed.

Construction of entanglement-assisted quantum MDS codes

Entanglement-assisted quantum error correction codes (EAQEECs) can improve the reliability of transmission, which make use of pre-existing entanglement between two parties. In this paper, we construct two classes of entanglement-assisted quantum maximum distance separable (EAQMDS) codes via generalized Reed-Solomon (GRS) codes. Consequently, the results show that some of these EAQMDS codes have much larger minimum distance than ones available in the literature. Meanwhile, many of these EAQMDS codes are new in the sense that the parameters of these codes are not covered by the previously known ones. ()

On the Asymptotic Capacity of X-Secure T-Private Information Retrieval With Graph-Based Replicated Storage

The problem of private information retrieval with graph-based replicated storage was recently introduced by Raviv, Tamo and Yaakobi. Its capacity remains open in almost all cases. In this work the asymptotic (large number of messages) capacity of this problem is studied along with its generalizations to include arbitrary T -privacy and X-security constraints, where the privacy of the user must be protected against any set of up to T colluding servers and the security of the stored data must be protected against any set of up to X colluding servers. A general achievable scheme for arbitrary storage patterns is presented that achieves the rate (ρ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">min</sub> -X -T )/N, where N is the total number of servers, and each message is replicated at least ρ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">min</sub> times. Notably, the scheme makes use of a special structure inspired by dual Generalized Reed Solomon (GRS) codes. A general converse is also presented. The two bounds are shown to match for many settings, including symmetric storage patterns. Finally, the asymptotic capacity is fully characterized for the case without security constraints (X = 0) for arbitrary storage patterns provided that each message is replicated no more than T + 2 times. As an example of this result, consider PIR with arbitrary graph based storage (T = 1, X = 0) where every message is replicated at exactly 3 servers. For this 3-replicated storage setting, the asymptotic capacity is equal to 2/ν <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> (G) where ν <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> (G) is the maximum size of a 2-matching in a storage graph G[V, E]. In this undirected graph, the vertices V correspond to the set of servers, and there is an edge uv ∈ E between vertices u, v only if a subset of messages is replicated at both servers u and v.

On Euclidean Hulls of MDS Codes

In this paper, we propose a mechanism for the construction of MDS codes with arbitrary dimensions of Euclidean hulls. Precisely, we construct (extended) generalized Reed-Solomon (GRS) codes with assigned dimensions of Euclidean hulls from self-orthogonal GRS codes. It turns out our constructions are more general than previous works on Euclidean hulls of (extended) GRS codes.

Some new entanglement-assisted quantum error-correcting MDS codes from generalized Reed–Solomon codes

Entanglement-assisted quantum maximum distance separable (MDS) codes form a significant class of quantum codes. By using generalized Reed–Solomon (GRS) codes and extended GRS codes, we construct some new classes of q-ary entanglement-assisted quantum error-correcting MDS codes. Most of these codes are new in the sense that their parameters are not covered by the codes available in the literature.

Euclidean and Hermitian Hulls of MDS Codes and Their Applications to EAQECCs

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.

New Entanglement-Assisted Quantum MDS Codes Derived From Generalized Reed-Solomon Codes

Entanglement-assisted quantum error-correcting codes (abbreviate to EAQECCs) expand the usual paradigm of quantum error correction by allowing two parties to make use of pre-shared entanglement. It is well-known that we can construct an EAQECC from arbitrary classical linear code. In this paper, we construct several classes of entanglement-assisted quantum MDS (EAQMDS) codes by utilizing generalized Reed-Solomon (GRS) codes. The main contribution of the paper is extend the code length of EAQMDS in the literature (Guo et al. 2019). Consequently, the results show that almost all of these EAQMDS codes are new in the sense that the parameters of these codes are not covered by the previously known ones.

Some New Constructions of Quantum MDS 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 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.

New LCD MDS codes constructed from generalized Reed–Solomon codes

Maximum distance separable codes with complementary duals (LCD MDS codes) are very important in coding theory and practice, and have attracted a lot of attention. In this paper, we focus on LCD MDS codes constructed from generalized Reed–Solomon (GRS) codes over a finite field with odd characteristic. We detail two constructions of new LCD MDS codes, using invertible matrices and the roots of three classes of polynomials, respectively.

Hermitian Self-Dual, MDS, and Generalized Reed–Solomon Codes

Both self-dual codes and maximum distance separable (MDS) codes have nice algebraic structures, theoretical significance, and practical implications. We present two classes of the Hermitian self-dual, MDS, and generalized Reed–Solomon (GRS) codes. Conversely, we prove that the Hermitian self-dual, MDS, and GRS codes must be above two classes for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${n =4}$ </tex-math></inline-formula> and conjecture that this result also holds for each even length <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$ {n}$ </tex-math></inline-formula> , <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$4\le {n}\le \text {q+1}$ </tex-math></inline-formula> .