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New MDS self-dual codes over finite fields of $r^2
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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 develop the existing theory and construct six new classes of MDS self-dual 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 and MDS almost self-dual codes are also constructed.
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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 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 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 .
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 MDS self-dual codes over large prime fields
- X Fang
- K Lebed
- H Liu
- J Luo
X. Fang, K. Lebed, H. Liu, J. Luo, New MDS self-dual codes over large
prime fields, Finite Fuelds Appl. 8 (4) (2002) 455-470
New constructions of MDS self-dual and selforthogonal codes via GRS codes
- Z Huang
- W Fang
- F Fu
Z. Huang, W. Fang, F. Fu, New constructions of MDS self-dual and selforthogonal codes via GRS codes. arXiv preprint arXiv:2103.11665 (2021).