• 제목/요약/키워드: self-dual code

검색결과 21건 처리시간 0.029초

SELF-DUAL CODES AND FIXED-POINT-FREE PERMUTATIONS OF ORDER 2

  • Kim, Hyun Jin
    • 대한수학회보
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    • 제51권4호
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    • pp.1175-1186
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    • 2014
  • We construct new binary optimal self-dual codes of length 50. We develop a construction method for binary self-dual codes with a fixed-point-free automorphism of order 2. Using this method, we find new binary optimal self-dual codes of length 52. From these codes, we obtain Lee-optimal self-dual codes over the ring $\mathbb{F}_2+u\mathbb{F}_2$ of lengths 25 and 26.

ON THE CONSTRUCTION OF MDS SELF-DUAL CODES OVER GALOIS RINGS

  • HAN, SUNGHYU
    • Journal of Applied and Pure Mathematics
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    • 제4권3_4호
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    • pp.211-219
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    • 2022
  • We study MDS(maximum distance separable) self-dual codes over Galois ring R = GR(2m, r). We prove that there exists an MDS self-dual code over R of length n if (n - 1) divides (2r - 1), and 2m divides n. We also provide the current state of the problem for the existence of MDS self-dual codes over Galois rings.

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

  • Shefali Gupta;Dinesh Udar
    • 대한수학회보
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    • 제60권3호
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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.

NONEXISTENCE OF SOME EXTREMAL SELF-DUAL CODES

  • Han, Sun-Ghyu;Lee, June-Bok
    • 대한수학회지
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    • 제43권6호
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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.

NOTES ON MDS SELF-DUAL CODES

  • Han, Sunghyu
    • Journal of applied mathematics & informatics
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    • 제30권5_6호
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    • pp.821-827
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    • 2012
  • In this paper, we prove that for all odd prime powers $q$ there exist MDS(maximum distance separable) self-dual codes over $\mathbb{F}_{q^2}$ for all even lengths up to $q+1$. Additionally, we prove that there exist MDS self-dual codes of length four over $\mathbb{F}_q$ for all $q$ > 2, $q{\neq}5$.

CONSTRUCTION OF SELF-DUAL CODES OVER F2 + uF2

  • Han, Sung-Hyu;Lee, Hei-Sook;Lee, Yoon-Jin
    • 대한수학회보
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    • 제49권1호
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    • pp.135-143
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    • 2012
  • We present two kinds of construction methods for self-dual codes over $\mathbb{F}_2+u\mathbb{F}_2$. Specially, the second construction (respectively, the first one) preserves the types of codes, that is, the constructed codes from Type II (respectively, Type IV) is also Type II (respectively, Type IV). Every Type II (respectively, Type IV) code over $\mathbb{F}_2+u\mathbb{F}_2$ of free rank larger than three (respectively, one) can be obtained via the second construction (respectively, the first one). Using these constructions, we update the information on self-dual codes over $\mathbb{F}_2+u\mathbb{F}_2$ of length 9 and 10, in terms of the highest minimum (Hamming, Lee, or Euclidean) weight and the number of inequivalent codes with the highest minimum weight.