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Construction of MDS Euclidean Self-Dual Codes via Multiple Subsets
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Abstract
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.
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Near maximum distance separable (NMDS) codes are important in finite geometry and coding theory. Self-dual codes are closely related to combinatorics, lattice theory, and have important application in cryptography. In this paper, we construct a class of q-ary linear codes and prove that they are either MDS or NMDS which depends on certain zero-sum condition. In the NMDS case, we provide an effective approach to construct NMDS self-dual codes which largely extend known parameters of such codes. In particular, we proved that for square q, almost q/8 NMDS self-dual q-ary codes can be constructed.
In this paper, we not only give the parity check matrix of the [1, 0]-twisted generalized Reed-Solomon (in short, TGRS) code, but also determine the weight distribution. Especially, we show that the [1, 0]-TGRS code is not GRS or EGRS. Furthermore, we present a sufficient and necessary condition for any punctured code of the [1, 0]-TGRS code to be self-orthogonal, and then construct several classes of self-dual or almost self-dual [1, 0]-TGRS codes. Finally, on the basis of these self-dual or almost self-dual [1, 0]-TGRS codes, we obtain some LCD [1, 0]-TGRS codes.
In this paper, we give a sufficient and necessary condition that a twisted Reed-Solomon (TRS) code is MDS. Then we give a sufficient and necessary condition that a twisted generalized Reed-Solomon (TGRS) code is self-dual. Moreover, we present some new explicit constructions of self-dual TGRS codes. These self-dual TGRS codes are MDS, Near-MDS, or 2-MDS and most of them are non-GRS.
Self-dual MDS and NMDS codes over finite fields are linear codes with significant combinatorial and cryptographic applications. In this paper, firstly, we investigate the duality properties of generalized twisted Reed-Solomon (abbreviated GTRS) codes in some special cases. Then, a new systematic approach is proposed to obtain Hermitian self-dual (+)-GTRS codes, and furthermore, necessary and sufficient conditions for a (+)-GTRS code to be Hermitian self-dual are presented. Finally, several classes of Hermitian self-dual MDS and NMDS codes are constructed with this method.
Let be a finite field of size q and the set of non-zero elements of . In this paper, we study a class of twisted generalized Reed–Solomon code generated by the following matrix where the evaluation set , scaling vector and . The minimum distance and dual code of will be determined. For the special case a sufficient and necessary condition for to be self-dual will be given. We will also show that the code is MDS or near-MDS. Moreover, a complete classification when the code is near-MDS or MDS will be presented.
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.
Self-dual maximum distance separable codes (self-dual MDS codes) and self-dual near MDS (NMDS) codes are very important in coding theory and practice. Thus, it is interesting to construct self-dual MDS or self-dual NMDS codes. In this paper, we not only give parity-check matrices of dual codes of twisted generalized Reed–Solomon codes (TGRS codes) but also present a sufficient and necessary condition of self-dual TGRS codes over with q an odd prime. Moreover, we construct several classes of self-dual MDS or self-dual NMDS codes from TGRS codes.
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.
Both self-dual codes and maximum distance separable (MDS) codes have nice algebraic structures, theoretical significant, and practical implications. We present two classes of Hermitian self-dual, MDS, and generalized Reed-Solomon (GRS) codes. Conversely, we prove that Hermitian self-dual, MDS, and GRS codes must be above two classes for n = 4 and conjecture that this result also holds for each even length n, 4≤n≤q+1.
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.
Secret sharing is an important topic in cryptography and has applications in information security. We use self-dual codes to construct secret-sharing schemes. We use combinatorial properties and invariant theory to understand the access structure of these secret-sharing schemes. We describe two techniques to determine the access structure of the scheme, the first arising from design properties in codes and the second from the Jacobi weight enumerator, and invariant theory.
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.
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.
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.
Recently, the construction of new MDS Euclidean self-dual codes has been widely investigated. In this paper, for square \begin{document} q \end{document}, we utilize generalized Reed-Solomon (GRS) codes and their extended codes to provide four generic families of \begin{document} q \end{document}-ary MDS Euclidean self-dual codes of lengths in the form \begin{document}\end{document}, where \begin{document} s \end{document} and \begin{document} t \end{document} range in some interval and \begin{document}\end{document}. In particular, for large square \begin{document} q \end{document}, our constructions take up a proportion of generally more than 34% in all the possible lengths of \begin{document} q \end{document}-ary MDS Euclidean self-dual codes, which is larger than the previous results. Moreover, two new families of MDS Euclidean self-orthogonal codes and two new families of MDS Euclidean almost self-dual codes are given similarly.
MDS self-dual codes over finite fields have attracted a lot of attention in recent years by their theoretical interests in coding theory and applications in cryptography and combinatorics. In this paper we present a series of MDS self-dual codes with new length by using generalized Reed-Solomon codes and extended generalized Reed-Solomon codes as the candidates of MDS codes and taking their evaluation sets as a union of cyclotomic classes. The conditions on such MDS codes being self-dual are expressed in terms of cyclotomic numbers.
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 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 self-dual NMDS codes. It turns out that, as long as 2|q and n is even with 4 ≤ n ≤ q + ⌊2 √q⌋ – 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+3 × (n + 3)2.
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 .
Orthogonal designs and generalized orthogonal designs are shown to be useful in construction of self-dual codes over large fields. For lengths 8, 10, 12 and 16, MDS self-dual codes over large fields are constructed from some orthogonal designs and generalized orthogonal designs. For length 14, some MDS self-dual codes over large fields are also constructed.
This paper is devoted to the study of self-dual codes arising from constacyclic codes. Necessary and sufficient conditions are given for the existence of Hermitian self-dual constacyclic codes over
F
q
2
of length
n
n
. As an application of these necessary and sufficient conditions, some conditions under which MDS Hermitian self-orthogonal and self-dual constacyclic codes exist are obtained.
Combinatorial designs have been used widely in the construction of self-dual codes. Recently a new method of constructing self-dual codes was established using orthogonal designs. This method has led to the construction of many new self-dual codes over small finite fields and rings. In this paper, we generalize this method by using generalized orthogonal designs, and we give another new method that creates and solves Diophantine equations over GF(p) in order to find suitable generator matrices for self-dual codes. We show that under the necessary conditions these methods can be applied as well to small and large fields. We apply these two methods to study self-dual codes over GF(31) and GF(37). Using these methods we obtain some new maximum distance separable self-dual codes of small orders.
In this paper we construct MDS Euclidean and Hermitian self-dual codes which are extended cyclic duadic codes or negacyclic
codes. We also construct Euclidean self-dual codes which are extended negacyclic codes. Based on these constructions, a large
number of new MDS self-dual codes are given with parameters for which self-dual codes were not previously known to exist.
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.
In this paper, we study self-dual codes over GF(p) where p=11,13,17,19,23 and 29. A classification of such codes for small lengths is given. The largest minimum weights of these codes are investigated. Many maximum distance separable self-dual codes are constructed.
New MDS or near-MDS self-dual codes
- T A Gulluver
- J.-L Kim
- Y Lee
T. A. Gulluver, J.-L. Kim and Y. Lee, New MDS or near-MDS self-dual codes,
IEEE Trans. Inf. Theory 54 (9) (2008) 4354-4360.