• Title, Summary, Keyword: codes

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SOME CLASSES OF REPEATED-ROOT CONSTACYCLIC CODES OVER 𝔽pm+u𝔽pm+u2𝔽pm

  • Liu, Xiusheng;Xu, Xiaofang
    • Journal of the Korean Mathematical Society
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    • v.51 no.4
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    • pp.853-866
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    • 2014
  • Constacyclic codes of length $p^s$ over $R=\mathbb{F}_{p^m}+u\mathbb{F}_{p^m}+u^2\mathbb{F}_{p^m}$ are precisely the ideals of the ring $\frac{R[x]}{<x^{p^s}-1>}$. In this paper, we investigate constacyclic codes of length $p^s$ over R. The units of the ring R are of the forms ${\gamma}$, ${\alpha}+u{\beta}$, ${\alpha}+u{\beta}+u^2{\gamma}$ and ${\alpha}+u^2{\gamma}$, where ${\alpha}$, ${\beta}$ and ${\gamma}$ are nonzero elements of $\mathbb{F}_{p^m}$. We obtain the structures and Hamming distances of all (${\alpha}+u{\beta}$)-constacyclic codes and (${\alpha}+u{\beta}+u^2{\gamma}$)-constacyclic codes of length $p^s$ over R. Furthermore, we classify all cyclic codes of length $p^s$ over R, and by using the ring isomorphism we characterize ${\gamma}$-constacyclic codes of length $p^s$ over R.

Implementation of systematic LT codes using VHDL (VHDL을 이용한 시스터메틱 LT 부호의 구현)

  • Zhang, Meixiang;Kim, Sooyoung;Chang, Jin Yeong;Kim, Won-Yong
    • Journal of Satellite, Information and Communications
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    • v.9 no.2
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    • pp.45-51
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    • 2014
  • Luby transform (LT) codes are a class of ratelss codes, and they have capability of generating infinite length of parities with a given information length. These rateless codes can be effectively utilized to provide broadcasting and multicasting services where each user is in a different channel condition. For this reason, there have been a number of researches on the application of rateless codes for satellite systems. In this paper, by considering the current research status on rateless codes, we present VHLD implementation results of LT codes, for future hardware implementation for satellite systems. The results demonstrated in this paper can be utilized as a basic information on efficient utilization of rateless codes in the future satellite systems.

IR-RBT Codes: A New Scheme of Regenerating Codes for Tolerating Node and Intra-node Failures in Distributed Storage Systems

  • Bian, Jianchao;Luo, Shoushan;Li, Wei;Zha, Yaxing;Yang, Yixian
    • KSII Transactions on Internet and Information Systems (TIIS)
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    • v.13 no.10
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    • pp.5058-5077
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    • 2019
  • Traditional regenerating codes are designed to tolerate node failures with optimal bandwidth overhead. However, there are many types of partial failures inside the node, such as latent sector failures. Recently, proposed regenerating codes can also repair intra-node failures with node-level redundancy but incur significant bandwidth and I/O overhead. In this paper, we construct a new scheme of regenerating codes, called IR-RBT codes, which employs intra-node redundancy to tolerate intra-node failures and serve as the help data for other nodes during the repair operation. We propose 2 algorithms for assigning the intra-node redundancy and RBT-Helpers according to the failure probability of each node, which can flexibly adjust the helping relationship between nodes to address changes in the actual situation. We demonstrate that the IR-RBT codes improve the bandwidth and I/O efficiency during intra-node failure repair over traditional regenerating codes but sacrifice the storage efficiency.

ADDITIVE SELF-DUAL CODES OVER FIELDS OF EVEN ORDER

  • Dougherty, Steven T.;Kim, Jon-Lark;Lee, Nari
    • Bulletin of the Korean Mathematical Society
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    • v.55 no.2
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    • pp.341-357
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    • 2018
  • We examine various dualities over the fields of even orders, giving new dualities for additive codes. We relate the MacWilliams relations and the duals of ${\mathbb{F}}_{2^{2s}}$ codes for these various dualities. We study self-dual codes with respect to these dualities and prove that any subgroup of order $2^s$ of the additive group is a self-dual code with respect to some duality.

THE q-ADIC LIFTINGS OF CODES OVER FINITE FIELDS

  • Park, Young Ho
    • Korean Journal of Mathematics
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    • v.26 no.3
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    • pp.537-544
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    • 2018
  • There is a standard construction of lifting cyclic codes over the prime finite field ${\mathbb{Z}}_p$ to the rings ${\mathbb{Z}}_{p^e}$ and to the ring of p-adic integers. We generalize this construction for arbitrary finite fields. This will naturally enable us to lift codes over finite fields ${\mathbb{F}}_{p^r}$ to codes over Galois rings GR($p^e$, r). We give concrete examples with all of the lifts.

ON THE EXTREMAL TYPE I BINARY SELF-DUAL CODES WITH NEAR-MINIMAL SHADOW

  • HAN, SUNGHYU
    • Journal of applied mathematics & informatics
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    • v.37 no.1_2
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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($m{\geq}87$).

NOTES ON MDS SELF-DUAL CODES

  • Han, Sunghyu
    • Journal of applied mathematics & informatics
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    • v.30 no.5_6
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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, $q{\neq}5$.

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

  • Kim, Hyun Jin
    • Bulletin of the Korean Mathematical Society
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    • v.51 no.4
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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.