• Title/Summary/Keyword: Linear codes

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Design of Quasi-Cyclic Low-Density Parity Check Codes with Large Girth

  • Jing, Long-Jiang;Lin, Jing-Li;Zhu, Wei-Le
    • ETRI Journal
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    • v.29 no.3
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    • pp.381-389
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    • 2007
  • In this paper we propose a graph-theoretic method based on linear congruence for constructing low-density parity check (LDPC) codes. In this method, we design a connection graph with three kinds of special paths to ensure that the Tanner graph of the parity check matrix mapped from the connection graph is without short cycles. The new construction method results in a class of (3, ${\rho}$)-regular quasi-cyclic LDPC codes with a girth of 12. Based on the structure of the parity check matrix, the lower bound on the minimum distance of the codes is found. The simulation studies of several proposed LDPC codes demonstrate powerful bit-error-rate performance with iterative decoding in additive white Gaussian noise channels.

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On the ${\kappa}$-Error Linear Complexity of $p^m$-Periodic Binary Sequences and Its Applications to Binary Cyclic Codes ($p^m$-주기 이진 수열의 ${\kappa}$-오류 선형복잡도와 이진 순환 부호에의 응용)

  • Han Yun-Kyoung;Yang Kyeong-Cheol
    • The Journal of Korean Institute of Communications and Information Sciences
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    • v.31 no.9C
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    • pp.846-852
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    • 2006
  • The ${\kappa}$-error linear complexity is a ky measure of the stability of the sequences used in the areas of communication systems, stream ciphers in cryptology and so on. This paper introduces an efficient algorithm to determine the ${\kappa}$-error linear complexity and the corresponding error vectors of $p^m$-periodic binary sequences, where : is a prime and 2 is a primitive root modulo $p^2$. We also give a new sense about the ${\kappa}$-error linear complexity in viewpoint of coding theory instead of cryptographic results. We present an efficient algorithm for decoding binary cyclic codes of length $p^m$ and derive key properties of the minimum distance of these codes.

NONEXISTENCE OF SOME EXTREMAL SELF-DUAL CODES

  • Han, Sun-Ghyu;Lee, June-Bok
    • Journal of the Korean Mathematical Society
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    • v.43 no.6
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    • pp.1357-1369
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    • 2006
  • It is known that if C is an [24m + 2l, 12m + l, d] selfdual binary linear code with $0{\leq}l<11,\;then\;d{\leq}4m+4$. We present a sufficient condition for the nonexistence of extremal selfdual binary linear codes with d=4m+4,l=1,2,3,5. From the sufficient condition, we calculate m's which correspond to the nonexistence of some extremal self-dual binary linear codes. In particular, we prove that there are infinitely many such m's. We also give similar results for additive self-dual codes over GF(4) of length n=6m+1.

The Mutual Information for Bit-Linear Linear-Dispersion Codes (BLLD 부호의 Mutual Information)

  • Jin, Xiang-Lan;Yang, Jae-Dong;Song, Kyoung-Young;No, Jong-Seon;Shin, Dong-Joon
    • The Journal of Korean Institute of Communications and Information Sciences
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    • v.32 no.10A
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    • pp.958-964
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    • 2007
  • In this paper, we derive the relationship between the bit error probability (BEP) of maximum a posteriori (MAP) bit detection and the bit minimum mean square error (MMSE), that is, the BEP is greater than a quarter of the bit USE and less than a half of the bit MMSE. By using this result, the lower and upper bounds of the derivative of the mutual information are derived from the BEP and the lower and upper bounds are easily obtained in the multiple-input multiple-output (MIMO) communication systems with the bit-linear linear-dispersion (BLLD) codes in the Gaussian channel.

AN UPPER BOUND ON THE NUMBER OF PARITY CHECKS FOR BURST ERROR DETECTION AND CORRECTION IN EUCLIDEAN CODES

  • Jain, Sapna;Lee, Ki-Suk
    • Journal of the Korean Mathematical Society
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    • v.46 no.5
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    • pp.967-977
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    • 2009
  • There are three standard weight functions on a linear code viz. Hamming weight, Lee weight, and Euclidean weight. Euclidean weight function is useful in connection with the lattice constructions [2] where the minimum norm of vectors in the lattice is related to the minimum Euclidean weight of the code. In this paper, we obtain an upper bound over the number of parity check digits for Euclidean weight codes detecting and correcting burst errors.

길이가 16인 Z$_4$위의 Preparata 부호는 연쇄조건을 만족하지 않는다

  • Kyeongcheol Yang;Dooroo Lim
    • Proceedings of the Korea Institutes of Information Security and Cryptology Conference
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    • 1996.11a
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    • pp.286-294
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    • 1996
  • In a remarkable paper 〔3〕, Hammons et al. showed that, when properly defined, the binary nonlinear Preparata code can be considered as the Gray map of a linear code eve. Z$_4$, the so-called Preparata code eve. Z$_4$. Recently, Yang and Helleseth 〔12〕 considered the generalized Hamming weights d$\_$r/(m) for Preparata codes of length 2$\^$m/ over Z$_4$ and exactly determined d$\_$r/, for r = 0.5,1.0,1.5,2,2.5 and 3.0. In particular, they completely determined d$\_$r/(m) for any r in the case of m $\leq$ 6. In this paper we show that the Preparata code of length 16 over Z$_4$ does not satisfy the chain condition.

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NEW RESULTS ON THE PSEUDOREDUNDANCY

  • Greferath, Marcus;Liu, Zihui;Wu, Xin-Wen;Zumbragel, Jens
    • Bulletin of the Korean Mathematical Society
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    • v.56 no.1
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    • pp.111-130
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    • 2019
  • The concepts of pseudocodeword and pseudoweight play a fundamental role in the finite-length analysis of LDPC codes. The pseudoredundancy of a binary linear code is defined as the minimum number of rows in a parity-check matrix such that the corresponding minimum pseudoweight equals its minimum Hamming distance. By using the value assignment of Chen and Kløve we present new results on the pseudocodeword redundancy of binary linear codes. In particular, we give several upper bounds on the pseudoredundancies of certain codes with repeated and added coordinates and of certain shortened subcodes. We also investigate several kinds of k-dimensional binary codes and compute their exact pseudocodeword redundancy.

AN ALTERED GROUP RING CONSTRUCTION OF THE [24, 12, 8] AND [48, 24, 12] TYPE II LINEAR BLOCK CODE

  • Shefali Gupta;Dinesh Udar
    • Bulletin of the Korean Mathematical Society
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    • v.60 no.3
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    • pp.829-844
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    • 2023
  • In this paper, we present a new construction for self-dual codes that uses the concept of double bordered construction, group rings, and reverse circulant matrices. Using groups of orders 2, 3, 4, and 5, and by applying the construction over the binary field and the ring F2 + uF2, we obtain extremal binary self-dual codes of various lengths: 12, 16, 20, 24, 32, 40, and 48. In particular, we show the significance of this new construction by constructing the unique Extended Binary Golay Code [24, 12, 8] and the unique Extended Quadratic Residue [48, 24, 12] Type II linear block code. Moreover, we strengthen the existing relationship between units and non-units with the self-dual codes presented in [10] by limiting the conditions given in the corollary. Additionally, we establish a relationship between idempotent and self-dual codes, which is done for the first time in the literature.

REPEATED LOW-DENSITY BURST ERROR DETECTING CODES

  • Dass, Bal Kishan;Verma, Rashmi
    • Journal of the Korean Mathematical Society
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    • v.48 no.3
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    • pp.475-486
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    • 2011
  • The paper deals with repeated low-density burst error detecting codes with a specied weight or less. Linear codes capable of detecting such errors have been studied. Further codes capable of correcting and simultaneously detecting such errors have also been dealt with. The paper obtains lower and upper bounds on the number of parity-check digits required for such codes. An example of such a code has also been provided.