• Journal of applied mathematics & informatics
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• 제37권1_2호
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• pp.85-95
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• 2019

In this paper, we define near-minimal shadow and study the existence problem of extremal Type I binary self-dual codes with near-minimal shadow. We prove that there is no such codes of length n = 24m + 2($m{\geq}0$), n = 24m + 4($m{\geq}9$), n = 24m + 6($m{\geq}21$), and n = 24m + 10($m{\geq}87$).

• Korean Journal of Mathematics
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• 제26권4호
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• pp.729-740
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• 2018

In this paper, we define near-minimal shadow and study the existence problem of extremal Type I additive self-dual codes over GF(4) with near-minimal shadow. We prove that there is no such codes if the code length n = 6m+1($m{\geq}0$), $n=6m+5(m{\geq}1)$.

• 대한수학회보
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• 제55권5호
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• pp.1371-1387
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• 2018

The classification of the self-dual codes over Galois rings GR(p, 2) and $GR(p^2,2)$ of length 4 is completed for all primes p up to equivalence in terms of automorphism group. We obtain all inequivalent classes and the number of each classes of self-dual codes for all primes.

• 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.

• 충청수학회지
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• 제23권4호
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• pp.893-902
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• 2010

In this article we study self-dual codes of length 6 over ${\mathbb{Z}}_m$. A classification of such codes for $m{\leq}24$ is given. Main tool for the classification is the new double cosets decomposition method given in the recent article of the author.

• 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(2^m, r). We prove that there exists an MDS self-dual code over R of length n if (n - 1) divides (2^r - 1), and 2^m divides n. We also provide the current state of the problem for the existence of MDS self-dual codes over Galois rings.

• 충청수학회지
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• 제31권2호
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• pp.269-280
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• 2018

We study self-dual codes over Galois rings using the building-up construction method. In the construction, the existence of an antiorthogonal matrix is very important. In this study, we examine the existence problem of an antiorthogonal matrix over Galois rings.

• 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.

• Korean Journal of Mathematics
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• 제28권2호
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• pp.285-294
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• 2020

We study the complementary information set codes (for short, CIS codes) over 𝔽₄. They are strongly connected to correlation-immune functions over 𝔽₄. Also the class of CIS codes includes the self-dual codes. We find a construction method of CIS codes over 𝔽₄ and a criterion for checking equivalence of CIS codes over 𝔽₄. We complete the classification of all inequivalent CIS codes of length up to 8 over 𝔽₄.

• Journal of Communications and Networks
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• 제15권1호
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• pp.71-76
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• 2013

A new class of quantum low-density parity-check (LDPC) codes whose parity-check matrices are dual-containing matrices constructed based on lines of Euclidean geometries (EGs) is presented. The parity-check matrices of our quantum codes contain one and only one 4-cycle in every two rows and have better distance properties. However, the classical parity-check matrix constructed from EGs does not satisfy the condition of dual-containing. In some parameter conditions, parts of the rows in the matrix maybe have not any nonzero element in common. Notably, we propose four families of fascinating structure according to changes in all the parameters, and the parity-check matrices are adopted to satisfy the requirement of dual-containing. Series of matrix properties are proved. Construction methods of the parity-check matrices with dual-containing property are given. The simulation results show that the quantum LDPC codes constructed by this method perform very well over the depolarizing channel when decoded with iterative decoding based on the sum-product algorithm. Also, the quantum codes constructed in this paper outperform other quantum codes based on EGs.