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New MDS Self-dual Codes Over Finite Field F r 2
Abstract
MDS self-dual codes have nice algebraic structures and are uniquely determined by lengths. Recently, the construction of MDS self-dual codes of new lengths has become an important and hot issue in coding theory. In this paper, we construct six new classes of MDS self-dual codes by using generalized Reed-Solomon (GRS for short) codes and extended GRS codes. Together with our constructions, the proportion of all known MDS self-dual codes relative to possible MDS self-dual codes generally exceed 57%. As far as we know, this is the largest known ratio. Moreover, some new families of MDS self-orthogonal codes are also constructed.
... Inspired by [7], Huang et al. further constructed some new families of MDS Euclidean self-dual codes in [15]. After this, Wan et al. [30] further generalized the above results and constructed six new classes of MDS Euclidean self-dual codes. And in [8], Fang et al. considered the union of two multiplicative subgroups with nonempty intersections and took their cosets as the evaluation sets. ...
... (4) In Table 2, we list the proportion of possible lengths relative to q 2 , where N is the number of all MDS Euclidean self-dual codes constructed in each reference and N 1 is the number of new lengths in our constructions. The constructions in [15] and [30] account for more than 34% and 38% of all possible MDS Euclidean self-dual codes. And in [8], the constructions can give more than 56% of all possible MDS Euclidean self-dual codes. ...
... 32 q = r 2 , r ≡ 3 (mod 4) n = s(r − 1) + t(r + 1), s odd, n = s q−1 a + t q−1 b , b and s are even, 2a | b(r + 1), 2b | a(r − 1), a ≡ 2 (mod 4), 1 ≤ s ≤ a gcd(a,b) , 1 ≤ t ≤ b gcd(a,b) [15] 34 q = r 2 , r ≡ 3 (mod 4) n = s q−1 a + t q−1 b , a even, (r+1)b 2a s 2 odd, 2a | b(r + 1), 2b | a(r − 1), b ≡ 2 (mod 4), 1 ≤ s ≤ a gcd(a,b) , 1 ≤ t ≤ b gcd(a,b) [15] 35 q = r 2 , r ≡ 1 (mod 4) n = s q−1 a + t q−1 b + 2, b even, s odd, 2a | b(r + 1), 2b | a(r − 1), a ≡ 2 (mod 4), 1 ≤ s ≤ a gcd(a,b) , 1 ≤ t ≤ b gcd(a,b) [15] 36 q = r 2 , r ≡ 3 (mod 4) n = s q−1 a + t q−1 b + 2, a even, 2a | b(r + 1), 2b | a(r − 1), b ≡ 2 (mod 4), (r+1)b 2a s 2 even, 1 ≤ s ≤ a gcd(a,b) , 1 ≤ t ≤ b gcd(a,b) [15] 37 q = r 2 , r ≡ 3 (mod 4) q = r 2 n = s q−1 e1 + t q−1 e2 , e 1 ≡ 2 l (mod 2 l+1 ), 2 l | e 2 , where l ≥ 2, 4 | (s − 1)(r + 1), 2e 2 | e 1 (r − 1), e 1 | e 2 (r + 1), 1 ≤ s ≤ e1 gcd(e1,e2) , 1 ≤ t ≤ e2 gcd(e1,e2) [30] 41 q = r 2 n = s q−1 e1 + t q−1 e2 + 1, e 1 ≡ 2 l (mod 2 l+1 ), 2 l | e 2 , where l ≥ 2, 4 | (s − 1)(r + 1), 2e 2 | e 1 (r − 1), e 1 | e 2 (r + 1), 1 ≤ s ≤ e1 gcd(e1,e2) , 1 ≤ t ≤ e2 gcd(e1,e2) [30] 42 q = r 2 n = s q−1 e1 + t q−1 e2 + 2, e 1 ≡ 2 l (mod 2 l+1 ), 2 l | e 2 , where l ≥ 2, 4 | (s − 1)(r + 1), 2e 2 | e 1 (r − 1), e 1 | e 2 (r + 1), 1 ≤ s ≤ e1 gcd(e1,e2) , 1 ≤ t ≤ e2 gcd(e1,e2) [30] 43 q = r 2 n = s q−1 e1 + t q−1 e2 , e 1 ≡ 2 l (mod 2 l+1 ), 2 l | e 2 , where l ≥ 2, r+1 2 ( te1 e2 + 1) even, 2e 2 | e 1 (r + 1), e 1 | e 2 (r − 1), 1 ≤ s ≤ e1 gcd(e1,e2) , 1 ≤ t ≤ e2 gcd(e1,e2) [30] n = s q−1 e1 + t q−1 e2 + 1, e 1 ≡ 2 l (mod 2 l+1 ), 2 l | e 2 , where l ≥ 2, r+1 2 ( te1 e2 + t) even, 4 | (s − 1)(r + 1), 2e 2 | e 1 (r + 1), e 1 | e 2 (r − 1), 1 ≤ s ≤ e1 gcd(e1,e2) , 1 ≤ t ≤ e2 gcd(e1,e2) [30] 45 q = r 2 n = s q−1 e1 + t q−1 e2 + 2, e 1 ≡ 2 l (mod 2 l+1 ), 2 l | e 2 , where l ≥ 2, r+1 2 ( te1 e2 + t) even, 4 | (s − 1)(r + 1), 2e 2 | e 1 (r + 1), e 1 | e 2 (r − 1), 1 ≤ s ≤ e1 gcd(e1,e2) , 1 ≤ t ≤ e2 gcd(e1,e2) [30] ...
MDS self-dual codes have good algebraic structure, and their parameters are completely determined by the code length. In recent years, the construction of MDS Euclidean self-dual codes with new lengths has become an important issue in coding theory. In this paper, we are committed to constructing new MDS Euclidean self-dual codes via generalized Reed-Solomon (GRS) codes and their extended (EGRS) codes. The main effort of our constructions is to find suitable subsets of finite fields as the evaluation sets, ensuring that the corresponding (extended) GRS codes are Euclidean self-dual. Firstly, we present a method for selecting evaluation sets from multiple intersecting subsets and provide a theorem to guarantee that the chosen evaluation sets meet the desired criteria. Secondly, based on this theorem, we construct six new classes of MDS Euclidean self-dual codes using the norm function, as well as the union of three multiplicity subgroups and their cosets respectively. Finally, in our constructions, the proportion of possible MDS Euclidean self-dual codes exceeds 85\%, which is much higher than previously reported results.
... Additionally, several classes of MDS self-dual codes are constructed via multiplicative subgroups or additive subgroups in finite fields ( [8], [9], [28]). In particular, a large class of MDS self-dual codes are derived in [30] via union of subsets over two multiplicative subgroups in finite fields. ...
In the realm of algebraic geometric (AG) codes, characterizing dual codes has long been a challenging task. In this paper we introduces a generalized criterion to characterize self-orthogonality of AG codes based on residues, drawing upon the rich algebraic structures of finite fields and the geometric properties of algebraic curves. We also present a generic construction of self-orthogonal AG codes from self-dual MDS codes. Using these approaches, we construct several families of self-dual and almost self-dual AG codes. These codes combine two merits: good performance as AG code whose parameters are close to the Singleton bound together with Euclidean (or Hermtian) self-dual/self-orthogonal property. Furthermore, some AG codes with Hermitian self-orthogonality can be applied to construct quantum codes with notably good parameters.
... In particular, a large class of MDS self-dual codes are derived in [26] via union of subsets over two multiplicative subgroups in finite fields. [24]. ...
In the field of algebraic geometric codes (AG codes), the characterization of dual codes has long been a challenging problem which relies on differentials. In this paper, we provide some descriptions for certain differentials utilizing algebraic structure of finite fields and geometric properties of algebraic curves. Moreover, we construct self-orthogonal and self-dual codes with parameters satisfying is close to n. Additionally, quantum codes with large minimum distance are also constructed.
... Naturally, there is a problem: how to construct self-dual codes? In recent year, authors studied self-dual (MDS) codes constructed by GRS codes [7,8,13,14,28,25,15,19] and TGRS codes [9,3,30,22,29]. For a 1-TGRS code to be self-dual, Huang et al. presented a condition that is both sufficient and necessary in [12]. ...
... (4) In [15], Fang et al. considered the union of two disjoint multiplicative subgroups and took their cosets as the evaluation sets. Then the conclusions in [15] was developed by Huang et al. [16] and Wan et al. [17]. ...
MDS self-dual codes have nice algebraic structures, theoretical significance and practical implications. In this paper, we present three classes of -ary Hermitian self-dual (extended) generalized Reed-Solomon codes with different code locators. Combining the results in Ball et al. (Designs, Codes and Cryptography, 89: 811-821, 2021), we show that if the code locators do not contain zero, -ary Hermitian self-dual (extended) GRS codes of length does not exist. Under certain conditions, we prove Conjecture 3.7 and Conjecture 3.13 proposed by Guo and Li et al. (IEEE Communications Letters, 25(4): 1062-1065, 2021).
Twisted Generalized Reed-Solomon (TGRS) Codes are a class of linear codes which contain MDS codes but are not equivalent to GRS code. In recent years, many scholars have studied the construction of MDS self-dual codes through TGRS code. Among these works, the number of twists is actually quite small, and an important step in showing a TGRS code is self-dual is to find its parity-check matrix. In this paper, we focus on the computation of parity-check matrices of TGRS codes with almost no restrictions on twist and hook vectors. By application of elementary row (column) operations in linear algebra, we present a unified but elementary approach to obtain the parity-check matrices of TGRS codes. This paper creates a foundation for further study of MDS self-dual TGRS codes on a much broader scope.
A linear code is called an MDS self-dual code if it is both an MDS code and a self-dual code with respect to the Euclidean inner product. The parameters of such codes are completely determined by the code length. In this paper, we consider new constructions of MDS self-dual codes via generalized Reed-Solomon (GRS) codes and their extended codes. The critical idea of our constructions is to choose suitable evaluation points such that the corresponding (extended) GRS codes are self-dual. The evaluation set of our constructions consists of a subgroup of finite fields and its cosets in a bigger subgroup. Four new families of MDS self-dual codes are then obtained. Moreover, by the Möbius action over finite fields, for any known self-dual GRS codes, we give a systematic way to construct new self-dual GRS codes with flexible evaluation points.
In this paper, we produce new classes of MDS self-dual codes via (extended) generalized Reed–Solomon codes over finite fields of odd characteristic. Among our constructions, there are many MDS self-dual codes with new parameters which have never been reported. When q is square of odd prime power, the total number of lengths of MDS self-dual codes over Fq presented in this paper is much more than those in all the previous results.
Due to the nice structures of maximum distance separable (MDS) codes and self-dual codes, it is worth studying MDS self-dual codes. It is easy to see that parameters of a MDS self-dual code are completely determined by the code length. The main problem in this topic is to determine existence of q-ary MDS self-dual codes of various lengths. The problem is completely solved when q is even. This paper focuses on the case that q is odd. We generalize the technique in [13] and construct several classes of MDS self-dual codes via generalized Reed-Solomon codes and extended generalized Reed-Solomon codes.
In this paper, we study a special kind of factorization of over with q a prime power when n=2p, with and p is a prime. Given such a q infinitely many such p's exist that admit q as a primitive root by the Artin conjecture in arithmetic progressions. This number theory conjecture is known to hold under GRH. We study the double (resp. four)-negacirculant codes over finite fields of co-index such n's, including the exact enumeration of the self-dual subclass, and a modified Varshamov-Gilbert bound on the relative distance of the codes it contains.
In this paper, we construct MDS Euclidean self-dual codes which are extended cyclic duadic codes. And we obtain many new MDS Euclidean self-dual codes. We also construct MDS Hermitian self-dual codes from generalized Reed-Solomon codes and constacyclic codes. And we give some results on Hermitian self-dual codes, which are the extended cyclic duadic codes.
Error-correcting codes and matroids have been widely used in the study of ordinary secret sharing schemes. In this paper, the connections between codes, matroids, and a special class of secret sharing schemes, namely, multiplicative linear secret sharing schemes (LSSSs), are studied. Such schemes are known to enable multiparty computation protocols secure against general (nonthreshold) adversaries. Two open problems related to the complexity of multiplicative LSSSs are considered in this paper. The first one deals with strongly multiplicative LSSSs. As opposed to the case of multiplicative LSSSs, it is not known whether there is an efficient method to transform an LSSS into a strongly multiplicative LSSS for the same access structure with a polynomial increase of the complexity. A property of strongly multiplicative LSSSs that could be useful in solving this problem is proved. Namely, using a suitable generalization of the well-known Berlekamp-Welch decoder, it is shown that all strongly multiplicative LSSSs enable efficient reconstruction of a shared secret in the presence of malicious faults. The second one is to characterize the access structures of ideal multiplicative LSSSs. Specifically, the considered open problem is to determine whether all self-dual vector space access structures are in this situation. By the aforementioned connection, this in fact constitutes an open problem about matroid theory, since it can be restated in terms of representability of identically self-dual matroids by self-dual codes. A new concept is introduced, the flat-partition, that provides a useful classification of identically self-dual matroids. Uniform identically self-dual matroids, which are known to be representable by self-dual codes, form one of the classes. It is proved that this property also holds for the family of matroids that, in a natural way, is the next class in the above classification: the identically self-dual bipartite matroids.
In this paper, for some fixed q, we construct some new classes of maximum distance separable (MDS) self-dual codes over Fq by using (extended) generalized Reed-Solomon codes. Our constructions utilize the additive and/or multiplicative structures of the finite field Fq. It reveals that by choosing suitable subsets of Fq, more MDS self-dual codes with new parameters can be found.
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.
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.
In this paper, two criterions of MDS Euclidean self-orthogonal codes are presented. New MDS Euclidean self-dual codes and self-orthogonal codes are constructed via the criterions. In particular, among our constructions, for large square q, about 1/8·q new MDS Euclidean (almost) self-dual codes over F
q
can be produced. Moreover, we can construct about 1/4· q new MDS Euclidean self-orthogonal codes with different even lengths n and dimension n/2-1.
MDS codes and self-dual codes are important families of classical codes in coding theory. Therefore, it is of interest to investigate MDS self-dual codes. The existence of MDS self-dual codes over finite field
is competely solved for
q
is even. In the literature, there are many known constructions of MDS self-dual codes for
q
is odd. In this paper, we present a unified approach on the existence of MDS self-dual codes with concise statements and simplified proof. It is illustrated that some of known results can also be stated in this framework. Furthermore, we can obtain some new MDS self-dual codes, especially for the case when
q
is not a square.
In this paper, we obtain some new results on the existence of MDS self-dual codes utilizing (extended) generalized Reed-Solomon codes over finite fields of odd characteristic. For finite field with odd characteristic and square cardinality, our results can produce more classes of MDS self-dual codes than previous works.
In this paper, we consider the problem for which lengths an 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 the present 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.
In this paper, we give algorithms and methods of construction of self-dual codes over finite fields using orthogonal matrices. Randomization in the orthogonal group, code extension and projection over a self-dual basis are the main tools. Some optimal, almost MDS, and MDS self-dual codes over both small and large finite fields are constructed. Moreover, over fifty MDS codes with new parameters are constructed. Comparisons with classical constructions are made.
In a distributed storage system, code symbols are dispersed across space in nodes or storage units as opposed to time. In settings such as that of a large data center, an important consideration is the efficient repair of a failed node. Efficient repair calls for erasure codes that in the face of node failure, are efficient in terms of minimizing the amount of repair data transferred over the network, the amount of data accessed at a helper node as well as the number of helper nodes contacted. Coding theory has evolved to handle these challenges by introducing two new classes of erasure codes, namely regenerating codes and locally recoverable codes as well as by coming up with novel ways to repair the ubiquitous Reed-Solomon code. This survey provides an overview of the efforts in this direction that have taken place over the past decade.
Both MDS 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 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. The current paper focuses on
the case where q is odd. We construct a few classes of new MDS self-dual code
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
and q is sufficiently large (say . Furthermore, we prove
that there exists a q-ary MDS self-dual code of length n if and n
satisfies one of the three conditions: (i) and n is even; (ii) q
is odd and is an odd divisor of ; (iii) and
n=2tr for any .
Fundamentals of Error Correcting Codes is an in-depth introduction to coding theory from both an engineering and mathematical viewpoint. As well as covering classical topics, there is much coverage of techniques which could only be found in specialist journals and book publications. Numerous exercises and examples and an accessible writing style make this a lucid and effective introduction to coding theory for advanced undergraduate and graduate students, researchers and engineers, whether approaching the subject from a mathematical, engineering or computer science background.
We consider the problem for which lengths a self-dual MDS code over Fq exists.We show that for q = 2m, there are self-dual MDS codes for all even lengths up to 2m. Furthermore, self-dual MDS codes of length q + 1 over Fq exist for all odd prime powers q. Additionally, we present some new self-dual MDS codes.
The first author constructed new extremal binary self-dual codes (IEEE Trans. Inform. Theory 47 (2001) 386–393) and new self-dual codes over GF(4) with the highest known minimum weights (IEEE Trans. Inform. Theory 47 (2001) 1575–1580). The method used was to build self-dual codes from a given self-dual code of a smaller length. In this paper, we develop a complete generalization of this method for the Euclidean and Hermitian self-dual codes over finite fields GF(q). Using this method we construct many Euclidean and Hermitian self-dual MDS (or near MDS) codes of length up to 12 over various finite fields GF(q), where q=8,9,16,25,32,41,49,53,64,81, and 128. Our results on the minimum weights of (near) MDS self-dual codes over large fields give a better bound than the Pless–Pierce bound obtained from a modified Gilbert–Varshamov bound.
The applicability of techniques in coding theory to problems in cryptography is illustrated by examples concerning secret-sharing schemes with tailored access priveleges, the design of perfect local randomizers, the construction of t-resilient functions, and the quantization of the nonlinearity of boolean functions. Some novel coding concepts, in particular the notions of minimal codewords in linear codes and of a partition of the space of n- tuples based on nonlinear systematic codes akin to the coset partition for linear codes, are shown to be necessary to treat the cryptographic problems considered. The concepts of dual codes and dual distance as well as the relation between codes and orthogonal arrays are seen to play a central role in these applications of coding theory to cryptography. 1 Introduction Coding theory, which had its inception in the late 1940's, is now generally regarded as a mature science. Cryptography on the other hand, at least in the public sector, is ...
New constructions of MDS self-dual and self-orthogonal codes via GRS codes
- huang
Construction of MDS self-dual codes from generalized Reed–Solomon codes
- wan
New constructions of MDS self-dual and self-orthogonal codes via GRS codes
- Z Huang
- W Fang
- F.-W Fu
MDS self-dual codes over large prime fields
- S Georgiou
- C Koukouvinos
Construction of MDS self-dual codes from generalized Reed–Solomon codes
- R Wan
- S Zhu
- J Li