Most bounds on the size of codes hold for any code, whether linear or nonlinear. In this paper we characterize a family of systematic. 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. 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. The nonexistence of some quaternary linear codes of. 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.
Classification of griesmer codes and dual transform. 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. In the cases when h 4, projective codes meeting the griesmer bound are known which are different from the codes described in this survey. Our constructions use a method of farrell 4 based on anticodes.
In this work we determine the automorphism group of the binary macdonald codes. Well known examples of codes meeting the griesmer bound are. In particular, we modify a binary simplex code by deleting certain columns from its generator matrix. Codes that attain one of these limits are called optimal. Affine blocking sets, threedimensional codes and the. All our lrcs have a small locality r 2 and r 3, moreover, most of our codes attain the griesmer bound. Conclusions all binary projective codes meeting the griesmer have been characterized. Shiromoto and storme gave a griesmer type bound for linear codes over. Mcdonald codes are an important class of maximum minimum distance errorcorrecting codes that meet the griesmer bound. A griesmer bound for linear codes over finite quasifrobenius rings. The distance optimality of these linear codes can be judged by the codetable of m.
Further, we derive and study properties of a class of two, three and four weight codes c n. A griesmer bound for linear codes over finite quasi. We prove that some of these codes are quasicyclic and attain the griesmer bound. All these codes contain the all1 codeword and are constructed from small multiple blocking sets in ag2,q.
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. Moreover, most of these codes are optimal with respect to the griesmer bound. Codes that attain the hamming bound are called perfect codes. The griesmer bound is often tighter than the plotkin bound, and its derivation leads to methods of constructing. In this article we give a griesmer type bound for linear codes over. The structure of all skew constacyclic codes is completely determined. Finally, some examples are given to illustrate the results. An improvement of the griesmer bound for some small. Binary and ternary lcd codes from projective spaces. Consequently, thanks to the griesmer bound, we get the following result. Classification of griesmer codes and dual transform classification of griesmer codes and dual transform bouyukliev, iliya g. On the algebraic parameters of convolutional codes with.
The codes which attain this bound are called maximum rank. On the construction of binary optimal lcd codes with short. An upper bound which applies specifically to b n, d is 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 minimum distance d. When gcde, m 1, we obtain a subclass of optimal codes which attain the griesmer bound. Important examples of linear codes meeting the griesmer bound are the simplex cod e hp03 section 1. There is also a very similar version for nonbinary 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.
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. 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. All the known codes that attain the bounds on the minimum. By the gray map, a family of oneweight ternary linear codes is obtained, whose parameters attain the plotkin bound and griesmer bound. 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. Levenshtein 823,824, levy 830, macdonald 868, mceliece and rumsey. It 16 oct 2019 onthekernelofz2slinearsimplexand macdonaldcodes. Further, these codes belong to the class of twoweight codes. Department of computer science and engineering, arizona state university, tempe 852875406, u. For q 3 the type 41 does not exist, since there are not enough. This work has been partially supported by the spanish mineco under grant tin201677918p aeifeder, ue, and by the catalan agaur under grant 2017sgr00463. 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. Goparaju and calderbank 7 presented binary cyclic lrcs with optimal dimension for distances 2. Asymptotically good 4quasi transitive algebraic geometry codes over prime elds, 2016.
The codes of the line type 43 all attain the parameters of the best known codes gra07. Binary locally repairable codes with high availability via. 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. Affine blocking sets, threedimensional codes and the griesmer bound affine blocking sets, threedimensional codes and the griesmer bound ball, simeon. Prominent examples of codes meeting the griesmer bound are the simplex code and the 5,6,11 3 golay code.
A griesmer bound for linear codes over finite quasifrobenius. In this article we give a griesmer type bound for linear codes over finite quasi. A z2slinear code is a binary code which is the gray map image of a z2sadditive code. Two nonlinear families of binary codes are obtained via the gray map. The motivation behind the construction of this code is the idea to achieve 100% error correction on the. Some interesting optimal binary codes are also obtained. Binary locally repairable codes with high availability via anticodes natalia silberstein technion and bgu, israel. Construction of a class of linear binary codes achieving the varshamov griesmer bound. Moreover, when gcde, m 2 or 3 we construct new infinite families of codes with at most four weights. Important examples of linear codes meeting the griesmer bound are the. The culture of critique hereafter, cofc was originally published in 1998 by praeger publishers, an imprint of greenwood publishing group, inc.
Binary lcd codes play a significant role for their advantage of low complexity for implementations against sidechannel attacks and fault injection attacks. Introduction locally repairable codes lrcs are a family. 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. Projective codes meeting the griesmer bound sciencedirect. Scientific opinion on dietary reference values for vitamin d. A weighted version of a result of hamada on minihypers and. Several new classes of linear codes with few weights. It provides binary lrcs which attain the cadambe mazumdar bound. 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. Blocktransitive algebraic geometry codes attaining the.
A characterization of codes meeting the griesmer bound. 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. 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. This bound is called the generalized singleton bound. Finite projective geometries play an important role in the study of these codes. A new class of nonbinary codes meeting the griesmer bound. On the kernel of z 2 slinear simplex and macdonald codes thanks. Many authors have investigated classes of codes meeting the griesmer bound see for example. Research visit of patrick govaerts at mta sztaki report on the stay from 2 november 2003 till 30 january 2004. In this paper, the problem of finding lcd codes over the binary field is discussed. Every nonzero linear combination of the top k u rows has weight 2k, any other nonzero linear combination has weight 2k12.
It is the generic term for ergocalciferol vitamin d 2 32 and cholecalciferol vitamin d 3. Using gauss sum, their weight distributions are provided. 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. In 8 garg has shown that the bound given by 5 is attained for k2. 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. 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. Macdonald in 3 derives an upper bound on minimum distance of a linear block code. This paper presents a construction for several families of optimal binary locally repairable codes lrcs with small locality 2 and 3. In some of these cases the codes have been characterized, but many cases still remain. It provides binary lrcs which attain the cadambemazumdar bound. The geometric approach to linear codes springerlink. Mcdonald, finite rings with identity, pure and applied mathematics, vol.
Asummary of robert jackalls moral mazes the theme of this work is that managers constantly adapt to the social environments of their organisations in order. A state space approach to periodic convolutional codes. Pdf on the covering radius of simplex codes researchgate. We also obtain a class of optimal ternary codes from twolee weight projective codes over r, which meet the griesmer bound. Optimal codes are often divisible, and dodunekov and manev showed that for a binary code meeting the griesmer bound, the power of 2. These results have applications in the construction of codes that attain t. The quaternary macdonald codes were discussed in 6 and the macdonald codes over finite field were discussed in 2. Examples include codes that have only one codeword, and codes that are the whole of. Publishers pdf, also known as version of record includes final page. The goal of this paper is to prove a stronger result than the above. We investigate the correspondence between families of linear codes with given properties via dual. Optimal binary locally repairable codes via anticodes.
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