In this paper we characterize a family of systematic. Further, these codes belong to the class of twoweight codes. This bound is called the generalized singleton bound. Consequently, thanks to the griesmer bound, we get the following result. Construction of a class of linear binary codes achieving the varshamov griesmer bound. All this indicates that the notion of cyclicity as introduced by piret and roos is the appropriate one for convolutional codes not only when it comes to the algebraic theory, but also for constructing good codes.
We construct families of threedimensional linear codes that attain the griesmer bound and give a nonexplicit construction of linear codes that are one away from the griesmer bound. Classification of griesmer codes and dual transform classification of griesmer codes and dual transform bouyukliev, iliya g. Two nonlinear families of binary codes are obtained via the gray map. On the kernel of z 2 slinear simplex and macdonald codes thanks. It provides binary lrcs which attain the cadambe mazumdar bound. It 16 oct 2019 onthekernelofz2slinearsimplexand macdonaldcodes. In this article we give a griesmer type bound for linear codes over finite quasi. Most bounds on the size of codes hold for any code, whether linear or nonlinear. The goal of this paper is to prove a stronger result than the above. By the gray map, a family of oneweight ternary linear codes is obtained, whose parameters attain the plotkin bound and griesmer bound. The griesmer bound is often tighter than the plotkin bound, and its derivation leads to methods of constructing.
Gbtc over prime elds based on the joint work mar a chara, ricardo podest a, ricardo toledano blocktransitive algebraic geometry codes attaining the tsfasmanvladutzink bound. Many authors have investigated classes of codes meeting the griesmer bound see for example. In the mathematics of coding theory, the griesmer bound, named after james hugo griesmer, is a bound on the length of linear binary codes of dimension k and minimum distance d. We also wish to mention a very recent approach for constructing good convolutional codes. Another example is given by the repeat codes, where each symbol of the message is repeated an odd fixed number of times to obtain a codeword where q 2. The codes of the line type 43 all attain the parameters of the best known codes gra07. Introduction locally repairable codes lrcs are a family. Classification of griesmer codes and dual transform. In this paper we give geometric proofs for several results on optimal linear codes and discuss generalizations that arise from the geometric approach.
When gcde, m 1, we obtain a subclass of optimal codes which attain the griesmer bound. The culture of critique hereafter, cofc was originally published in 1998 by praeger publishers, an imprint of greenwood publishing group, inc. Codes that attain one of these limits are called optimal. In the cases when h 4, projective codes meeting the griesmer bound are known which are different from the codes described in this survey. Shiromoto and storme gave a griesmer type bound for linear codes over. A griesmer bound for linear codes over finite quasifrobenius. Finite projective geometries play an important role in the study of these codes. Conclusions all binary projective codes meeting the griesmer have been characterized.
In this article we give a griesmer type bound for linear codes over. Every nonzero linear combination of the top k u rows has weight 2k, any other nonzero linear combination has weight 2k12. Our constructions use a method of farrell 4 based on anticodes. For q 3 the type 41 does not exist, since there are not enough. An improvement of the griesmer bound for some small. Several new classes of linear codes with few weights. Of course, the natural setting here is the correspondence between linear codes and projective multisets. In this work we determine the automorphism group of the binary macdonald codes.
Scientific opinion on dietary reference values for vitamin d. The torsion code, weight distribution, and gray image properties are studied. In the next section we give a nonexplicit construction of such a set s and ls which gives codes that are one away from the griesmer bound. We prove that some of these codes are quasicyclic and attain the griesmer bound. The distance optimality of these linear codes can be judged by the codetable of m. Binary lcd codes play a significant role for their advantage of low complexity for implementations against sidechannel attacks and fault injection attacks. This is one of several papers published in designs, codes and cryptography comprising the special issue on finite geometries this research is done as a part of the project finite geometries, coding theory and cryptography between the research foundation flanders fwo and the bulgarian academy of sciences. Finally, some examples are given to illustrate the results. Pdf on the covering radius of simplex codes researchgate.
Affine blocking sets, threedimensional codes and the. The structure of all skew constacyclic codes is completely determined. All these codes contain the all1 codeword and are constructed from small multiple blocking sets in ag2,q. Codes that attain the hamming bound are called perfect codes. This work has been partially supported by the spanish mineco under grant tin201677918p aeifeder, ue, and by the catalan agaur under grant 2017sgr00463. A characterization of codes meeting the griesmer bound. In the mathematics of coding theory, the griesmer bound, named after james hugo griesmer, is a bound on the length of linear binary codes of dimension k and. Further, we derive and study properties of a class of two, three and four weight codes c n. Optimal codes are often divisible, and dodunekov and manev showed that for a binary code meeting the griesmer bound, the power of 2. Affine blocking sets, threedimensional codes and the griesmer bound affine blocking sets, threedimensional codes and the griesmer bound ball, simeon. Projective codes meeting the griesmer bound sciencedirect. Levenshtein 823,824, levy 830, macdonald 868, mceliece and rumsey. Some interesting optimal binary codes are also obtained. Optimal binary locally repairable codes via anticodes.
Prominent examples of codes meeting the griesmer bound are the simplex code and the 5,6,11 3 golay code. Some codes attain the griesmer bound 5 prove some new facts about krawtchouk polynomials ilya dumer olga kapralova university of california spherically punctured biorthogonal codes acct 2012 4 17. Notably, the griesmer bound, holds only in the linear case. The geometric approach to such problems is based on the equivalence between qary linear codes with no coordinate identically zero and multisets of points in projective geometries over finite fields.
Well known examples of codes meeting the griesmer bound are. Important examples of linear codes meeting the griesmer bound are the simplex cod e hp03 section 1. Mcdonald, finite rings with identity, pure and applied mathematics, vol. A state space approach to periodic convolutional codes. In some of these cases the codes have been characterized, but many cases still remain. On the algebraic parameters of convolutional codes with. It provides binary lrcs which attain the cadambemazumdar bound. Binary locally repairable codes with high availability via anticodes natalia silberstein technion and bgu, israel. Optimal binary linear locally repairable codes with disjoint repair groups jingxue ma and gennian ge abstract in recent years, several classes of codes are introduced to provide some faulttolerance and guarantee system reliability in distributed storage systems, among which locally repairable codes lrcs for short play an important role.
The nonexistence of some quaternary linear codes of. Mcdonald codes are an important class of maximum minimum distance errorcorrecting codes that meet the griesmer bound. Macdonald in 3 derives an upper bound on minimum distance of a linear block code. An upper bound which applies specifically to b n, d is the griesmer bound. Publishers pdf, also known as version of record includes final page. A griesmer bound for linear codes over finite quasifrobenius rings. Binary locally repairable codes with high availability via. There is no author who has so positively impacted my life the way macdonald has, and i am forever grateful for the person who first introduced me to his works. Abstractwe construct families of threedimensional linear codes that attain the griesmer bound and give a nonexplicit construction of linear codes that are one away from the griesmer bound.
Examples include codes that have only one codeword, and codes that are the whole of. Binary and ternary lcd codes from projective spaces. In 8 garg has shown that the bound given by 5 is attained for k2. All the known codes that attain the bounds on the minimum. All our lrcs have a small locality r 2 and r 3, moreover, most of our codes attain the griesmer bound. Codes meeting the griesmer bound 129 it is important that s 1 in the above conjecture, since helleseth and van tilborg 1981 have constructed codes meeting the griesmer bound with s 1 for which the conditions of theorem 1.
A griesmer bound for linear codes over finite quasi. A new class of nonbinary codes meeting the griesmer bound. We also obtain a class of optimal ternary codes from twolee weight projective codes over r, which meet the griesmer bound. Blocktransitive algebraic geometry codes attaining the. A weighted version of a result of hamada on minihypers and. The thesis of the book is a difficult one indeeddifficult not only because it is difficult to establish, but also because it challenges many fundamental. On the construction of binary optimal lcd codes with short. There is also a very similar version for nonbinary codes. In this article we give a griesmer type bound for linear codes over finite quasifrobenius rings and consider linear codes over these rings meeting the bound. The quaternary macdonald codes were discussed in 6 and the macdonald codes over finite field were discussed in 2. On the extendability of quasidivisible griesmer arcs. Goparaju and calderbank 7 presented binary cyclic lrcs with optimal dimension for distances 2. Moreover, when gcde, m 2 or 3 we construct new infinite families of codes with at most four weights.
This paper presents a construction for several families of optimal binary locally repairable codes lrcs with small locality 2 and 3. The geometric approach to linear codes springerlink. A divisor of a linear code is an integer dividing the weights of all its words, and a code is called divisible if it has a divisor larger than 1 20. The macdonald codes were introduced by macdonald 12 for p 2 and in 15 see also 2 for any q, and are simplex codes punctured in a particular way. Department of computer science and engineering, arizona state university, tempe 852875406, u. In particular, we modify a binary simplex code by deleting certain columns from its generator matrix. Moreover, most of these codes are optimal with respect to the griesmer bound. The codes which attain this bound are called maximum rank. Asymptotically good 4quasi transitive algebraic geometry codes over prime elds, 2016. Using gauss sum, their weight distributions are provided.
The motivation behind the construction of this code is the idea to achieve 100% error correction on the. It is wellknown 21 that over su ciently large nite elds, there always exist convolutional codes that achieve this bound for any given set of parameters n. This is the model and inspiration every christian needs, and macdonald brings these things to our understanding so that we can fully relate them to our own lives. A z2slinear code is a binary code which is the gray map image of a z2sadditive code. Research visit of patrick govaerts at mta sztaki report on the stay from 2 november 2003 till 30 january 2004. Important examples of linear codes meeting the griesmer bound are the.
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