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When Does the extended code of an MDS code Remain MDS?

IEEE TRANSACTIONS ON INFORMATION THEORY 2025年第1期71卷 263-272页

作者： Wu, Yansheng Ding, Cunsheng Chen, Tingfang Nanjing Univ Posts & Telecommun Sch Comp Sci Nanjing 210023 Peoples R China Hong Kong Univ Sci & Technol Dept Comp Sci & Engn Hong Kong Peoples R China

For a given linear code C of length n over GF(q) and a nonzero vector u in GF(q)(n) , Sun, Ding and Chen defined an extended linear code C(u) of C , which is a generalisation of the classical extended code C(-1) of C and called the second kind of an extended code of C (see Finite Fields Appl., vol. 96, 102401, 2024 and Discrete Math., vol. 347, no. 9, 114080, 2024). They developed some general theory of the extended codes C(u) and studied the extended codes C(u) of several families of linear codes, including cyclic codes, projective two-weight codes, nonbinary Hamming codes, and a family of reversible MDS cyclic codes. The objective of this paper is to investigate the extended codes C(u) of MDS codes C over finite fields. The main result of this paper is that the extended code C(u) of an MDS [n,k] code C remains MDS if and only if the covering radius rho(C-perpendicular to)=k and the vector u is a deep hole of the dual code C-perpendicular to . As applications of this main result, an equivalent statement of MDS Conjecture is presented, the extended codes of the GRS codes and extended GRS codes are investigated, and the covering radii and some deep holes of several families of MDS codes are also determined.

关键词： codes codecs Generators Vectors Linear codes Parity check codes Computer science Reed-Solomon codes Telecommunications Sun extended code MDS code covering radius deep hole GRS code extended GRS code

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Simulation of subcooled flow instability for high flux research reactors using the extended code ATHLET

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NUCLEAR ENGINEERING AND DESIGN 2001年第2期207卷 163-180页

作者： Hainoun, A Schaffrath, A Atom Energy Commis Syria Dept Phys Damascus Syria Forschungszentrum Rossendorf EV Inst Sicherheitsforsch D-01314 Dresden Germany

Covering the wide range of reactor safety analysis of power reactors, consisting of leak and transients, the thermohydraulic code ATHLET is being developed by the Gesellschaft for Anlagen- und Reaktorsicherheit (GRS-Society for Plant and Reactor Safety) (Lerchel, G., Austregesilo, H., 1998. ATHLET,ET Mode 1.2 Cycle A, User's Manual, GRS-p-1/Vol. 1, Rev. 1, GRS). In order to extend the code's range of application to the safety analysis of research reactors, a model was developed and implemented permitting a description of the steam formation in the subcooled boiling regime (Hainoun, A., 1994. Modellierung des unterkuhlten Siedens in ATHLET und Anwendung in wassergekuhlten Forschungsreaktoren, D 294 Diss. Univ. Bochum, Jul-2961). Considering the specific features of high flux research reactors given by both high heat flux and high flow velocity, the model of void condensation in subcooled flow has been extended and a new correlation of critical heat Aux (CHF) is implemented. To validate the extended program, the Thermal-Hydraulic Test Loop (THTL) of Oak Ridge National Laboratory (ORNL) was modeled with ATHLET and an extensive series of experiments concerning the onset of thermohydraulic flow instability (OFI) in subcooled boiling regime were calculated. The comparison between experiments and ATHLET-postcalculation shows that the extended code can accurately simulate the thermohydraulic conditions of Row instability in a wide range of heat flux up to 15 MW m(-2) and inlet flow velocity up to 20 m s(-1) The thermohydrauhc design limit characterized by the mass flux, at which the flow just becomes unstable (OFI), has been predicted in very good agreement with the experiment. However the calculated pressure drop at OFI is overestimated by a maximum deviation of about 25%. The calculated exit bulk temperature of subcooled coolant and the maximum wall temperature at OFI show a maximum deviation from experiment of 12 and 7%,, respectively. (C) 2001 Elsevier Science B.V. All

关键词： Mass flux high flux Heat flux subcooled boiling simulating thermohydraulics OAK RIDGE NATIONAL LABORATORY flow instability Research reactors maximum error Undercooling Safety Analysis Current velocity extended code

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Relations between the local weight distributions of a linear block code, its extended code, and its even weight subcode

Relations between the local weight distributions of a linear...

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IEEE International Symposium on Information Theory

作者： Yasunaga, K Fujiwara, T Osaka Univ Grad Sch Informat Sci & Technol Suita Osaka 5650871 Japan

ISBN: (纸本)0780391500

Relations between the local weight distributions of a binary linear code, its extended code, and its even weight subcode are presented. In particular, for a code of which the extended code is transitive invariant and contains only codewords with weight multiples of four, the local weight distribution can be obtained from that of the extended code. Using the relations, the local weight distributions of the (127, k) primitive BCH codes for k <= 50, the (127, 64) punctured third-order Reed-Muller, and their even weight subcodes are obtained from the local weight distribution of the (128, k) extended primitive BCH codes for k <= 50 and the (128,64) third-order Reed-Muller code. We also show an approach to improve an algorithm for computing the local weight distribution proposed before.

关键词： BCH codes Reed-Muller codes binary codes block codes linear codes (127, 64) punctured third-order Reed-Muller (127, k) extended primitive BCH codes (128, 64) third-order Reed-Muller code binary linear code even weight subcode extended code linear block c linear codes extended code BCH codes Reed-Muller codes binary code Block codes subcode linear block

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Recognition of Partially Occluded Objects Based on Segmentation by Median Filtering of extended Direction code

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Wuhan University Journal of Natural Sciences 1997年第4期2卷 53-56页

作者： Wang Yanping Larmagnac Jean-Pierre Yuan Jie Zhao Hengzhuo College of Electronic Information Wuhan University Wuhan China Dept. Appl. Physics I. U. T. Saint-Etinenne Cedex 2 France

The image contour is segmented into lines, arcs and smooth curves by median filtering of extended direction code. Based on this segmentation, a set of new local invariant features are proposed to recognize partially occluded objects, which is more reasonable compared with conventional corner features. The matching results of some typical examples shows that these features are robust ,effective in recognition.

关键词： pattern recognition Freeman's code extended code median filtering corner

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The extended codes of a Family of Reversible MDS Cyclic codes

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IEEE TRANSACTIONS ON INFORMATION THEORY 2024年第7期70卷 4808-4822页

作者： Sun, Zhonghua Ding, Cunsheng Hefei Univ Technol Sch Math Hefei 230601 Anhui Peoples R China Hong Kong Univ Sci & Technol Dept Comp Sci & Engn Hong Kong Peoples R China

A linear code with parameters [n, k, n-k+1] is called a maximum distance separable (MDS for short) code. A linear code with parameters [n, k, n-k] is said to be almost maximum distance separable (AMDS for short). A linear code is said to be near maximum distance separable (NMDS for short) if both the code and its dual are AMDS. MDS codes are very important in both theory and practice. There is a classical construction of a [q+1, 2u-1, q-2u+3] MDS code for each u with 1 <= u <= [q+1 / 2], which is a reversible and cyclic code. The objective of this paper is to study the extended codes of this family of MDS codes. Two families of MDS codes and several families of NMDS codes are obtained. The NMDS codes have applications in finite geometry, cryptography and distributed and cloud data storage systems. The weight distributions of some of the extended codes are determined.

关键词： codes Linear codes Generators Symbols Sun codecs Projective geometry Cyclic code extended code linear code locally recoverable code MDS code subfield code

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The extended codes of some linear codes

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FINITE FIELDS AND THEIR APPLICATIONS 2024年 96卷

作者： Sun, Zhonghua Ding, Cunsheng Chen, Tingfang Hefei Univ Technol Sch Math Hefei 230601 Peoples R China Hong Kong Univ Sci & Technol Dept Comp Sci & Engn Kowloon Clear Water Bay Hong Kong Peoples R China

The classical way of extending an [ n, k, d ] linear code C is to add an overall parity -check coordinate to each codeword of the linear code C . This extended code, denoted by C ( - 1 ) and called the standardly extended code of C , is a linear code with parameters [ n +1, k, d ], where d = d or d = d +1. This is one of the two extending techniques for linear codes in the literature. The standardly extended codes of some families of binary linear codes have been studied to some extent. However, not much is known about the standardly extended codes of nonbinary codes. For example, the minimum distances of the standardly extended codes of the nonbinary Hamming codes remain open for over 70 years. The first objective of this paper is to introduce the nonstandardly extended codes of a linear code and develop some general theory for this type of extended linear codes. The second objective is to study this type of extended codes of a number of families of linear codes, including cyclic codes and nonbinary Hamming codes. Four families of distance -optimal or dimension -optimal linear codes are obtained with this extending technique. The parameters of certain extended codes of many families of linear codes are settled in this paper. (c) 2024 Elsevier Inc. All rights reserved.

关键词： extended code Hamming code Constacyclic code Cyclic code Linear code

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The extended codes of projective two-weight codes

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DISCRETE MATHEMATICS 2024年第9期347卷

作者： Sun, Zhonghua Ding, Cunsheng Chen, Tingfang Hefei Univ Technol Sch Math Hefei 230601 Peoples R China Hong Kong Univ Sci & Technol Dept Comp Sci & Engn Clear Water Bay Hong Kong Peoples R China

A kind of extended codes of linear codes C with respect to a vector u , denoted by C(u), was introduced by Sun, Ding and Chen (2024) [58]. In this paper, we investigate the weight distribution of the extended codes C(u) of projective two-weight codes Cover finite fields. As a by-product, several families of linear codes with a few weights are presented, and six infinite families of distance-optimal linear codes are constructed in this paper. (c) 2024 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

关键词： extended code Projective two-weight code Constacyclic code Ovoid code Denniston maximal arc code

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Construction of punctured and extended quantum codes over GF(2)

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Science China(Information Sciences) 2013年第3期56卷 149-159页

作者： XIAO FangYing CHEN HanWu XING MeiJu LIU ZhiHao School of Computer Science and Engineering Southeast University

On the basis of elementary transformation,we propose a new method for constructing a class of pure quantum codes [[n-i,2k-n+i,d-i]]2 and [[n+1,2k-n-1,d+1]]2 from a class of classical linear codes [n,k,d]2 that contain their dual *** construction process was based on the elementary algebra;the error-correcting performance of the quantum codes was analyzed according to the relationship between the parity-check matrix and the minimum distance of the classical linear codes;the encoding and decoding networks were constructed based on the *** proposed method is simple,straightforward and easy to implement using a computer and other hardware *** theoretical results showed that the method is practical for the construction of a class of quantum codes.

关键词： Calderbank-Shor-Steane code punctured code extended code Hamming code check matrix

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On the binary codes with parameters of triply-shortened 1-perfect codes

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DESIGNS codeS AND CRYPTOGRAPHY 2012年第3期64卷 275-283页

作者： Krotov, Denis S. Novosibirsk State Univ Dept Mech & Math Novosibirsk 630090 Russia Sobolev Inst Math Novosibirsk 630090 Russia

We study properties of binary codes with parameters close to the parameters of 1-perfect codes. An arbitrary binary (n = 2 (m) - 3, 2 (n-m-1), 4) code C, i.e., a code with parameters of a triply-shortened extended Hamming code, is a cell of an equitable partition of the n-cube into six cells. An arbitrary binary (n = 2 (m) - 4, 2 (n-m) , 3) code D, i.e., a code with parameters of a triply-shortened Hamming code, is a cell of an equitable family (but not a partition) with six cells. As a corollary, the codes C and D are completely semiregular;i.e., the weight distribution of such codes depends only on the minimal and maximal codeword weights and the code parameters. Moreover, if D is self-complementary, then it is completely regular. As an intermediate result, we prove, in terms of distance distributions, a general criterion for a partition of the vertices of a graph (from rather general class of graphs, including the distance-regular graphs) to be equitable.

关键词： Coding theory Hamming code extended code 1-perfect code Triply-shortened 1-perfect code Equitable partition Perfect coloring Weight distribution Distance distribution

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Correcting Deletions Using Linear and Cyclic codes

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IEEE TRANSACTIONS ON INFORMATION THEORY 2010年第10期56卷 5223-5234页

作者： Abdel-Ghaffar, Khaled A. S. Ferreira, Hendrik C. Cheng, Ling Univ Calif Davis Dept Elect & Comp Engn Davis CA 95616 USA Univ Johannesburg Dept Elect & Elect Engn Sci ZA-2006 Auckland Pk South Africa

Linear and cyclic codes are typically used to combat substitution errors. However, synchronization errors, associated with the deletion and insertion of symbols, can cause severe performance degradation unless the coding scheme possesses the capability to recover from such errors. It is shown that linear codes of rate greater than 1/2 cannot correct deletion or insertion errors but there are linear codes of rate 1/2 that can correct these errors. Although cyclic codes, except for repetition codes, cannot correct deletion or insertion errors, two approaches are investigated to yield codes, based on cyclic codes, that can correct these errors. In the first approach, it is shown that a binary or nonbinary cyclic code of rate at most 1/3 or 1/2, respectively, can be extended by one symbol to make it capable of correcting synchronization errors. In the second approach, a cyclic code of rate at most 1/2 is expurgated by appropriately deleting codewords such that the expurgated code is capable of correcting synchronization errors. It is shown that deleting codewords costs at most two information bits if the code is binary and one information symbol if the code is nonbinary.

关键词： Cyclic code deletion expurgated code extended code insertion linear code substitution error synchronization error

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