154,929 research outputs found

    Maximum Weight Spectrum Codes

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    In the recent work \cite{shi18}, a combinatorial problem concerning linear codes over a finite field \F_q was introduced. In that work the authors studied the weight set of an [n,k]q[n,k]_q linear code, that is the set of non-zero distinct Hamming weights, showing that its cardinality is upper bounded by qkβˆ’1qβˆ’1\frac{q^k-1}{q-1}. They showed that this bound was sharp in the case q=2 q=2 , and in the case k=2 k=2 . They conjectured that the bound is sharp for every prime power q q and every positive integer k k . In this work quickly establish the truth of this conjecture. We provide two proofs, each employing different construction techniques. The first relies on the geometric view of linear codes as systems of projective points. The second approach is purely algebraic. We establish some lower bounds on the length of codes that satisfy the conjecture, and the length of the new codes constructed here are discussed.Comment: 19 page

    Relaxation Bounds on the Minimum Pseudo-Weight of Linear Block Codes

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    Just as the Hamming weight spectrum of a linear block code sheds light on the performance of a maximum likelihood decoder, the pseudo-weight spectrum provides insight into the performance of a linear programming decoder. Using properties of polyhedral cones, we find the pseudo-weight spectrum of some short codes. We also present two general lower bounds on the minimum pseudo-weight. The first bound is based on the column weight of the parity-check matrix. The second bound is computed by solving an optimization problem. In some cases, this bound is more tractable to compute than previously known bounds and thus can be applied to longer codes.Comment: To appear in the proceedings of the 2005 IEEE International Symposium on Information Theory, Adelaide, Australia, September 4-9, 200

    MWS and FWS Codes for Coordinate-Wise Weight Functions

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    A combinatorial problem concerning the maximum size of the (hamming) weight set of an [n,k]q[n,k]_q linear code was recently introduced. Codes attaining the established upper bound are the Maximum Weight Spectrum (MWS) codes. Those [n,k]q[n,k]_q codes with the same weight set as Fqn \mathbb{F}_q^n are called Full Weight Spectrum (FWS) codes. FWS codes are necessarily ``short", whereas MWS codes are necessarily ``long". For fixed k,q k,q the values of n n for which an [n,k]q [n,k]_q -FWS code exists are completely determined, but the determination of the minimum length M(H,k,q) M(H,k,q) of an [n,k]q [n,k]_q -MWS code remains an open problem. The current work broadens discussion first to general coordinate-wise weight functions, and then specifically to the Lee weight and a Manhattan like weight. In the general case we provide bounds on n n for which an FWS code exists, and bounds on n n for which an MWS code exists. When specializing to the Lee or to the Manhattan setting we are able to completely determine the parameters of FWS codes. As with the Hamming case, we are able to provide an upper bound on M(L,k,q) M(\mathcal{L},k,q) (the minimum length of Lee MWS codes), and pose the determination of M(L,k,q) M(\mathcal{L},k,q) as an open problem. On the other hand, with respect to the Manhattan weight we completely determine the parameters of MWS codes.Comment: 17 page

    Decoding and constructions of codes in rank and Hamming metric

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    As coding theory plays an important role in data transmission, decoding algorithms for new families of error correction codes are of great interest. This dissertation is dedicated to the decoding algorithms for new families of maximum rank distance (MRD) codes including additive generalized twisted Gabidulin (AGTG) codes and Trombetti-Zhou (TZ) codes, decoding algorithm for Gabidulin codes beyond half the minimum distance and also encoding and decoding algorithms for some new optimal rank metric codes with restrictions. We propose an interpolation-based decoding algorithm to decode AGTG codes where the decoding problem is reduced to the problem of solving a projective polynomial equation of the form q(x) = xqu+1 +bx+a = 0 for a,b ∈ Fqm. We investigate the zeros of q(x) when gcd(u,m)=1 and proposed a deterministic algorithm to solve a linearized polynomial equation which has a close connection to the zeros of q(x). An efficient polynomial-time decoding algorithm is proposed for TZ codes. The interpolation-based decoding approach transforms the decoding problem of TZ codes to the problem of solving a quadratic polynomial equation. Two new communication models are defined and using our models we manage to decode Gabidulin codes beyond half the minimum distance by one unit. Our models also allow us to improve the complexity for decoding GTG and AGTG codes. Besides working on MRD codes, we also work on restricted optimal rank metric codes including symmetric, alternating and Hermitian rank metric codes. Both encoding and decoding algorithms for these optimal families are proposed. In all the decoding algorithms presented in this thesis, the properties of Dickson matrix and the BM algorithm play crucial roles. We also touch two problems in Hamming metric. For the first problem, some cryptographic properties of Welch permutation polynomial are investigated and we use these properties to determine the weight distribution of a binary linear codes with few weights. For the second one, we introduce two new subfamilies for maximum weight spectrum codes with respect to their weight distribution and then we investigate their properties.Doktorgradsavhandlin

    Constructing Linear Encoders with Good Spectra

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    Linear encoders with good joint spectra are suitable candidates for optimal lossless joint source-channel coding (JSCC), where the joint spectrum is a variant of the input-output complete weight distribution and is considered good if it is close to the average joint spectrum of all linear encoders (of the same coding rate). In spite of their existence, little is known on how to construct such encoders in practice. This paper is devoted to their construction. In particular, two families of linear encoders are presented and proved to have good joint spectra. The first family is derived from Gabidulin codes, a class of maximum-rank-distance codes. The second family is constructed using a serial concatenation of an encoder of a low-density parity-check code (as outer encoder) with a low-density generator matrix encoder (as inner encoder). In addition, criteria for good linear encoders are defined for three coding applications: lossless source coding, channel coding, and lossless JSCC. In the framework of the code-spectrum approach, these three scenarios correspond to the problems of constructing linear encoders with good kernel spectra, good image spectra, and good joint spectra, respectively. Good joint spectra imply both good kernel spectra and good image spectra, and for every linear encoder having a good kernel (resp., image) spectrum, it is proved that there exists a linear encoder not only with the same kernel (resp., image) but also with a good joint spectrum. Thus a good joint spectrum is the most important feature of a linear encoder.Comment: v5.5.5, no. 201408271350, 40 pages, 3 figures, extended version of the paper to be published in IEEE Transactions on Information Theor
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