• Title, Summary, Keyword: 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.

Synchronous All-Optical Code-Division Multiple-Access Local-Area Networks with Symmetric Codes

  • Lam, Pham Manh;Praepanichawat, Chanikarn
    • Proceedings of the IEEK Conference
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    • pp.1109-1112
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    • 2002
  • A non-coherent synchronous all-optical code-division multiple-access (CDMA) network is proposed. In this network, symmetric codes derived from prime sequence codes are used. We present the construction of symmetric codes and show that the pseudo-orthogonality of the new codes is the same as that of the original prime-sequence codes while the cardinality of the new codes is larger than that of the prime sequence codes and the modified prime codes in the same field GF(p). Therefore, an optical CDMA LAN using symmetric codes can have a larger number of potential subscribers. The new codes allow designing fully programmable serial all-optical transmitter and receiver suitable for low-loss, high-capacity, optical CDMA LANs. It is also shown that compared to systems using modified prime codes the proposed system can achieve better BER performance for low received chip optical power.

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PRIMITIVE IDEMPOTENTS IN THE RING F4[x]/〈xpn-1〉 AND CYCLOTOMIC Q CODES

  • Batra, Sudhir;Mathur, Rekha
    • Bulletin of the Korean Mathematical Society
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    • v.55 no.3
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    • pp.971-997
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    • 2018
  • The parity of cyclotomic numbers of order 2, 4 and 6 associated with 4-cyclotomic cosets modulo an odd prime p are obtained. Hence the explicit expressions of primitive idempotents of minimal cyclic codes of length $p^n$, $n{\geq}1$ over the quaternary field $F_4$ are obtained. These codes are observed to be subcodes of Q codes of length $p^n$. Some orthogonal properties of these subcodes are discussed. The minimal cyclic codes of length 17 and 43 are also discussed and it is observed that the minimal cyclic codes of length 17 are two weight codes. Further, it is shown that a Q code of prime length is always cyclotomic like a binary duadic code and it seems that there are infinitely many prime lengths for which cyclotomic Q codes of order 6 exist.

QUANTUM CODES WITH IMPROVED MINIMUM DISTANCE

  • Kolotoglu, Emre;Sari, Mustafa
    • Bulletin of the Korean Mathematical Society
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    • v.56 no.3
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    • pp.609-619
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    • 2019
  • The methods for constructing quantum codes is not always sufficient by itself. Also, the constructed quantum codes as in the classical coding theory have to enjoy a quality of its parameters that play a very important role in recovering data efficiently. In a very recent study quantum construction and examples of quantum codes over a finite field of order q are presented by La Garcia in [14]. Being inspired by La Garcia's the paper, here we extend the results over a finite field with $q^2$ elements by studying necessary and sufficient conditions for constructions quantum codes over this field. We determine a criteria for the existence of $q^2$-cyclotomic cosets containing at least three elements and present a construction method for quantum maximum-distance separable (MDS) codes. Moreover, we derive a way to construct quantum codes and show that this construction method leads to quantum codes with better parameters than the ones in [14].

The Construction and Viterbi Decoding of New (2k, k, l) Convolutional Codes

  • Peng, Wanquan;Zhang, Chengchang
    • Journal of Information Processing Systems
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    • v.10 no.1
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    • pp.69-80
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    • 2014
  • The free distance of (n, k, l) convolutional codes has some connection with the memory length, which depends on not only l but also on k. To efficiently obtain a large memory length, we have constructed a new class of (2k, k, l) convolutional codes by (2k, k) block codes and (2, 1, l) convolutional codes, and its encoder and generation function are also given in this paper. With the help of some matrix modules, we designed a single structure Viterbi decoder with a parallel capability, obtained a unified and efficient decoding model for (2k, k, l) convolutional codes, and then give a description of the decoding process in detail. By observing the survivor path memory in a matrix viewer, and testing the role of the max module, we implemented a simulation with (2k, k, l) convolutional codes. The results show that many of them are better than conventional (2, 1, l) convolutional codes.

ON GENERALIZATIONS OF SKEW QUASI-CYCLIC CODES

  • Bedir, Sumeyra;Gursoy, Fatmanur;Siap, Irfan
    • Bulletin of the Korean Mathematical Society
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    • v.57 no.2
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    • pp.459-479
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    • 2020
  • In the last two decades, codes over noncommutative rings have been one of the main trends in coding theory. Due to the fact that noncommutativity brings many challenging problems in its nature, still there are many open problems to be addressed. In 2015, generator polynomial matrices and parity-check polynomial matrices of generalized quasi-cyclic (GQC) codes were investigated by Matsui. We extended these results to the noncommutative case. Exploring the dual structures of skew constacyclic codes, we present a direct way of obtaining parity-check polynomials of skew multi-twisted codes in terms of their generators. Further, we lay out the algebraic structures of skew multipolycyclic codes and their duals and we give some examples to illustrate the theorems.

Performance of Successive-Cancellation List Decoding of Extended-Minimum Distance Polar Codes (최소거리가 확장된 극 부호의 연속 제거 리스트 복호 성능)

  • Ryu, Daehyeon;Kim, Jae Yoel;Kim, Jong-Hwan;Kim, Sang-Hyo
    • The Journal of Korean Institute of Communications and Information Sciences
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    • v.38C no.1
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    • pp.109-117
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    • 2013
  • Polar codes are the first provable error correcting code achieving the symmetric channel capacity in a wide case of binary input discrete memoryless channel(BI-DMC). However, finite length polar codes have an error floor problem with successive-cancellation list(SCL) decoder. From previous works, we can solve this problem by concatenating CRC(Cyclic Redundancy Check) codes. In this paper we propose to make polar codes having extended-minimum distance from original polar codes without outer codes using correlation with generate matrix of polar codes and that of RM(Reed-Muller) codes. And we compare performance of proposed polar codes with that of polar codes concatenating CRC codes.

CIS CODES OVER ��4

  • Kim, Hyun Jin
    • Korean Journal of Mathematics
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    • v.28 no.2
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    • pp.285-294
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    • 2020
  • We study the complementary information set codes (for short, CIS codes) over ��4. They are strongly connected to correlation-immune functions over ��4. Also the class of CIS codes includes the self-dual codes. We find a construction method of CIS codes over ��4 and a criterion for checking equivalence of CIS codes over ��4. We complete the classification of all inequivalent CIS codes of length up to 8 over ��4.

m-ADIC RESIDUE CODES OVER Fq[v]/(v2 - v) AND DNA CODES

  • Kuruz, Ferhat;Oztas, Elif Segah;Siap, Irfan
    • Bulletin of the Korean Mathematical Society
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    • v.55 no.3
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    • pp.921-935
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    • 2018
  • In this study we determine the structure of m-adic residue codes over the non-chain ring $F_q[v]/(v^2-v)$ and present some promising examples of such codes that have optimal parameters with respect to Griesmer Bound. Further, we show that the generators of m-adic residue codes serve as a natural and suitable application for generating reversible DNA codes via a special automorphism and sets over $F_{4^{2k}}[v]/(v^2-v)$.

SKEW CONSTACYCLIC CODES OVER FINITE COMMUTATIVE SEMI-SIMPLE RINGS

  • Dinh, Hai Q.;Nguyen, Bac Trong;Sriboonchitta, Songsak
    • Bulletin of the Korean Mathematical Society
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    • v.56 no.2
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    • pp.419-437
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    • 2019
  • This paper investigates skew ${\Theta}-{\lambda}$-constacyclic codes over $R=F_0{\oplus}F_1{\oplus}{\cdots}{\oplus}F_{k-1}$, where $F{_i}^{\prime}s$ are finite fields. The structures of skew ${\lambda}$-constacyclic codes over finite commutative semi-simple rings and their duals are provided. Moreover, skew ${\lambda}$-constacyclic codes of arbitrary length are studied under a new definition. We also show that a skew cyclic code of arbitrary length over finite commutative semi-simple rings is equivalent to either a cyclic code over R or a quasi-cyclic code over R.