750 research outputs found

    Projective divisible binary codes

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    For which positive integers n,k,rn,k,r does there exist a linear [n,k][n,k] code CC over Fq\mathbb{F}_q with all codeword weights divisible by qrq^r and such that the columns of a generating matrix of CC are projectively distinct? The motivation for studying this problem comes from the theory of partial spreads, or subspace codes with the highest possible minimum distance, since the set of holes of a partial spread of rr-flats in PG(v1,Fq)\operatorname{PG}(v-1,\mathbb{F}_q) corresponds to a qrq^r-divisible code with kvk\leq v. In this paper we provide an introduction to this problem and report on new results for q=2q=2.Comment: 10 pages, 3 table

    Coding Theory and Algebraic Combinatorics

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    This chapter introduces and elaborates on the fruitful interplay of coding theory and algebraic combinatorics, with most of the focus on the interaction of codes with combinatorial designs, finite geometries, simple groups, sphere packings, kissing numbers, lattices, and association schemes. In particular, special interest is devoted to the relationship between codes and combinatorial designs. We describe and recapitulate important results in the development of the state of the art. In addition, we give illustrative examples and constructions, and highlight recent advances. Finally, we provide a collection of significant open problems and challenges concerning future research.Comment: 33 pages; handbook chapter, to appear in: "Selected Topics in Information and Coding Theory", ed. by I. Woungang et al., World Scientific, Singapore, 201

    Binary Cyclic Pearson Codes

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    The phenomena of unknown gain or offset on communication systems and modern storages such as optical data storage and non-volatile memory (flash) becomes a serious problem. This problem can be handled by Pearson distance applied to the detector because it offers immunity to gain and offset mismatch. This distance can only be used for a specific set of codewords, called Pearson codes. An interesting example of Pearson code can be found in T-constrained code class. In this paper, we present binary 2-constrained codes with cyclic property. The construction of this code is adopted from cyclic codes, but it cannot be considered as cyclic codes.Fenomena gain atau offset yang tidak terduga pada sistem komunikasi dan media penyimpan data modern seperti media penyimpanan berjenis optik (CD) dan memori non-volatile (flash) merupakan gangguan yang serius. Permasalahan ini dapat ditangani dengan mengaplikasikan jarak Pearson pada detektor error pada sistem tersebut karena jarak Pearson menawarkan kekebalan terhadap gain dan offset yang tidak menentu. Jarak ini hanya dapat digunakan pada suatu himpunan codewords tertentu, yaitu himpunan Pearson/kode Pearson. Salah satu contoh kode Pearson dapat ditemukan di kelas kode T-constranied. Dalam paper ini, diusulkan kode 2-constrained biner dengan sifat siklis. Konstruksi kode ini diadopsi dari konstruksi pada kode siklis, akan tetapi kode yang dihasilkan tidak dapat dipandang sebagai kode siklis

    The Subfield Codes of Some Few-Weight Linear Codes

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    Subfield codes of linear codes over finite fields have recently received a lot of attention, as some of these codes are optimal and have applications in secrete sharing, authentication codes and association schemes. In this paper, the qq-ary subfield codes Cˉf,g(q)\bar{C}_{f,g}^{(q)} of six different families of linear codes Cˉf,g\bar{C}_{f,g} are presented, respectively. The parameters and weight distribution of the subfield codes and their punctured codes Cˉf,g(q)\bar{C}_{f,g}^{(q)} are explicitly determined. The parameters of the duals of these codes are also studied. Some of the resultant qq-ary codes Cˉf,g(q),\bar{C}_{f,g}^{(q)}, Cˉf,g(q)\bar{C}_{f,g}^{(q)} and their dual codes are optimal and some have the best known parameters. The parameters and weight enumerators of the first two families of linear codes Cˉf,g\bar{C}_{f,g} are also settled, among which the first family is an optimal two-weight linear code meeting the Griesmer bound, and the dual codes of these two families are almost MDS codes. As a byproduct of this paper, a family of [24m2,2m+1,24m3][2^{4m-2},2m+1,2^{4m-3}] quaternary Hermitian self-dual code are obtained with m2m \geq 2. As an application, several infinite families of 2-designs and 3-designs are also constructed with three families of linear codes of this paper.Comment: arXiv admin note: text overlap with arXiv:1804.06003, arXiv:2207.07262 by other author

    Distance-regular graphs

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    This is a survey of distance-regular graphs. We present an introduction to distance-regular graphs for the reader who is unfamiliar with the subject, and then give an overview of some developments in the area of distance-regular graphs since the monograph 'BCN' [Brouwer, A.E., Cohen, A.M., Neumaier, A., Distance-Regular Graphs, Springer-Verlag, Berlin, 1989] was written.Comment: 156 page

    Recent progress on weight distributions of cyclic codes over finite fields

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    Cyclic codes are an interesting type of linear codes and have wide applications in communication and storage systems due to their efficient encoding and decoding algorithms. In coding theory it is often desirable to know the weight distribution of a cyclic code to estimate the error correcting capability and error probability. In this paper, we present the recent progress on the weight distributions of cyclic codes over finite fields, which had been determined by exponential sums. The cyclic codes with few weights which are very useful are discussed and their existence conditions are listed. Furthermore, we discuss the more general case of constacyclic codes and give some equivalences to characterize their weight distributions
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