• Title/Summary/Keyword: linear codes

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WEIGHT ENUMERATORS OF TWO CLASSES OF LINEAR CODES

  • Ahn, Jaehyun;Ka, Yeonseok
    • Journal of the Chungcheong Mathematical Society
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    • v.33 no.1
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    • pp.43-56
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    • 2020
  • Recently, linear codes constructed from defining sets have been studied widely and determined their complete weight enumerators and weight enumerators. In this paper, we obtain complete weight enumerators of linear codes and weight enumerators of linear codes. These codes have at most three weight linear codes. As application, we show that these codes can be used in secret sharing schemes and authentication codes.

OPTIMAL LINEAR CODES OVER ℤm

  • Dougherty, Steven T.;Gulliver, T. Aaron;Park, Young-Ho;Wong, John N.C.
    • Journal of the Korean Mathematical Society
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    • v.44 no.5
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    • pp.1139-1162
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    • 2007
  • We examine the main linear coding theory problem and study the structure of optimal linear codes over the ring ${\mathbb{Z}}_m$. We derive bounds on the maximum Hamming weight of these codes. We give bounds on the best linear codes over ${\mathbb{Z}}_8$ and ${\mathbb{Z}}_9$ of lengths up to 6. We determine the minimum distances of optimal linear codes over ${\mathbb{Z}}_4$ for lengths up to 7. Some examples of optimal codes are given.

ON LCD CODES OVER FINITE CHAIN RINGS

  • Durgun, Yilmaz
    • Bulletin of the Korean Mathematical Society
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    • v.57 no.1
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    • pp.37-50
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    • 2020
  • Linear complementary dual (LCD) codes are linear codes that intersect with their dual trivially. LCD cyclic codes have been known as reversible cyclic codes that had applications in data storage. Due to a newly discovered application in cryptography, interest in LCD codes has increased again. Although LCD codes over finite fields have been extensively studied so far, little work has been done on LCD codes over chain rings. In this paper, we are interested in structure of LCD codes over chain rings. We show that LCD codes over chain rings are free codes. We provide some necessary and sufficient conditions for an LCD code C over finite chain rings in terms of projections of linear codes. We also showed the existence of asymptotically good LCD codes over finite chain rings.

CONSTRUCTION OF TWO- OR THREE-WEIGHT BINARY LINEAR CODES FROM VASIL'EV CODES

  • Hyun, Jong Yoon;Kim, Jaeseon
    • Journal of the Korean Mathematical Society
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    • v.58 no.1
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    • pp.29-44
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    • 2021
  • The set D of column vectors of a generator matrix of a linear code is called a defining set of the linear code. In this paper we consider the problem of constructing few-weight (mainly two- or three-weight) linear codes from defining sets. It can be easily seen that we obtain an one-weight code when we take a defining set to be the nonzero codewords of a linear code. Therefore we have to choose a defining set from a non-linear code to obtain two- or three-weight codes, and we face the problem that the constructed code contains many weights. To overcome this difficulty, we employ the linear codes of the following form: Let D be a subset of ��2n, and W (resp. V ) be a subspace of ��2 (resp. ��2n). We define the linear code ��D(W; V ) with defining set D and restricted to W, V by $${\mathcal{C}}_D(W;V )=\{(s+u{\cdot}x)_{x{\in}D^{\ast}}|s{\in}W,u{\in}V\}$$. We obtain two- or three-weight codes by taking D to be a Vasil'ev code of length n = 2m - 1(m ≥ 3) and a suitable choices of W. We do the same job for D being the complement of a Vasil'ev code. The constructed few-weight codes share some nice properties. Some of them are optimal in the sense that they attain either the Griesmer bound or the Grey-Rankin bound. Most of them are minimal codes which, in turn, have an application in secret sharing schemes. Finally we obtain an infinite family of minimal codes for which the sufficient condition of Ashikhmin and Barg does not hold.

LINEAR AND NON-LINEAR LOOP-TRANSVERSAL CODES IN ERROR-CORRECTION AND GRAPH DOMINATION

  • Dagli, Mehmet;Im, Bokhee;Smith, Jonathan D.H.
    • Bulletin of the Korean Mathematical Society
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    • v.57 no.2
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    • pp.295-309
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    • 2020
  • Loop transversal codes take an alternative approach to the theory of error-correcting codes, placing emphasis on the set of errors that are to be corrected. Hitherto, the loop transversal code method has been restricted to linear codes. The goal of the current paper is to extend the conceptual framework of loop transversal codes to admit nonlinear codes. We present a natural example of this nonlinearity among perfect single-error correcting codes that exhibit efficient domination in a circulant graph, and contrast it with linear codes in a similar context.

Nonlinear Product Codes and Their Low Complexity Iterative Decoding

  • Kim, Hae-Sik;Markarian, Garik;Da Rocha, Valdemar C. Jr.
    • ETRI Journal
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    • v.32 no.4
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    • pp.588-595
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    • 2010
  • This paper proposes encoding and decoding for nonlinear product codes and investigates the performance of nonlinear product codes. The proposed nonlinear product codes are constructed as N-dimensional product codes where the constituent codes are nonlinear binary codes derived from the linear codes over higher order alphabets, for example, Preparata or Kerdock codes. The performance and the complexity of the proposed construction are evaluated using the well-known nonlinear Nordstrom-Robinson code, which is presented in the generalized array code format with a low complexity trellis. The proposed construction shows the additional coding gain, reduced error floor, and lower implementation complexity. The (64, 24, 12) nonlinear binary product code has an effective gain of about 2.5 dB and 1 dB gain at a BER of $10^{-6}$ when compared to the (64, 15, 16) linear product code and the (64, 24, 10) linear product code, respectively. The (256, 64, 36) nonlinear binary product code composed of two Nordstrom-Robinson codes has an effective gain of about 0.7 dB at a BER of $10^{-5}$ when compared to the (256, 64, 25) linear product code composed of two (16, 8, 5) quasi-cyclic codes.

A Two-Step Screening Algorithm to Solve Linear Error Equations for Blind Identification of Block Codes Based on Binary Galois Field

  • Liu, Qian;Zhang, Hao;Yu, Peidong;Wang, Gang;Qiu, Zhaoyang
    • KSII Transactions on Internet and Information Systems (TIIS)
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    • v.15 no.9
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    • pp.3458-3481
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    • 2021
  • Existing methods for blind identification of linear block codes without a candidate set are mainly built on the Gauss elimination process. However, the fault tolerance will fall short when the intercepted bit error rate (BER) is too high. To address this issue, we apply the reverse algebra approach and propose a novel "two-step-screening" algorithm by solving the linear error equations on the binary Galois field, or GF(2). In the first step, a recursive matrix partition is implemented to solve the system linear error equations where the coefficient matrix is constructed by the full codewords which come from the intercepted noisy bitstream. This process is repeated to derive all those possible parity-checks. In the second step, a check matrix constructed by the intercepted codewords is applied to find the correct parity-checks out of all possible parity-checks solutions. This novel "two-step-screening" algorithm can be used in different codes like Hamming codes, BCH codes, LDPC codes, and quasi-cyclic LDPC codes. The simulation results have shown that it can highly improve the fault tolerance ability compared to the existing Gauss elimination process-based algorithms.

ON THE CONSTRUCTION OF OPTIMAL LINEAR CODES OF DIMENSION FOUR

  • Atsuya Kato;Tatsuya Maruta;Keita Nomura
    • Bulletin of the Korean Mathematical Society
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    • v.60 no.5
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    • pp.1237-1252
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    • 2023
  • A fundamental problem in coding theory is to find nq(k, d), the minimum length n for which an [n, k, d]q code exists. We show that some q-divisible optimal linear codes of dimension 4 over 𝔽q, which are not of Belov type, can be constructed geometrically using hyperbolic quadrics in PG(3, q). We also construct some new linear codes over 𝔽q with q = 7, 8, which determine n7(4, d) for 31 values of d and n8(4, d) for 40 values of d.