For a linear code and a linear subcode, the relative dimension/length profile (RDLP), inverse relative dimension/length profile (IRDLP) and relative length/dimension profile (RLDP) are three character sequences, which...
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ISBN:
(纸本)9781612843063
For a linear code and a linear subcode, the relative dimension/length profile (RDLP), inverse relative dimension/length profile (IRDLP) and relative length/dimension profile (RLDP) are three character sequences, which have been applied to the wiretap channel of type II with illegitimate parties and extended in the wiretap network II for the secrecy control of linear network coding. They also provide useful information to analyze the trellis complexity of a linear code. These concepts are two-code generalizations of the DLP, IDLP and LDP proposed by Forney, respectively. The Singleton bounds on RDLP, IRDLP and RLDP also extend Forney's bounds. In this paper, we introduce new relations with respect to RDLP and IRDLP. We show that the Singleton bounds on RDLP, IRDLP and RLDP can be derived directly from these relations. Some interesting duality properties of RDLP and IRDLP are also given.
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 ...
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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 les 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 les 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
Network coding is a promising generalization of routing which allows a node to generate output messages by encoding its received messages. An important scenario where network coding offers unique advantages is a multi...
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Network coding is a promising generalization of routing which allows a node to generate output messages by encoding its received messages. An important scenario where network coding offers unique advantages is a multicast network where a source node generates messages and multiple receivers collect the messages. In a multicast network, linear network codes are preferred due to its sufficiency and simplicity. In this paper, we propose a method to transform the linear coding problem to a graph theory problem. With the help of hypergraphs, we model the linear codes by constructing a pseudo-dual graph of the multicast network. A valid linear code is equal to a cover in the pseudo-dual graph satisfying some constraints. By iterative refinements, an eligible cover can be found in polynomial time. Moreover, this method can be readily applied to many minimum network coding problems as well.
Fault-tolerant systolic arrays for matrix-vector multiplication are proposed. A novel method called the linear arithmetic code technique is introduced which can detect and correct n errors caused by a single faulty mo...
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Fault-tolerant systolic arrays for matrix-vector multiplication are proposed. A novel method called the linear arithmetic code technique is introduced which can detect and correct n errors caused by a single faulty module in a linear systolic array. This approach can be extended to other matrix operations.< >
linear codes with few weights have wide applications in consumer electronics, data storage system and secret sharing. In this paper, by virtue of planar functions, several infinite families of l-weight linear codes ov...
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linear codes with few weights have wide applications in consumer electronics, data storage system and secret sharing. In this paper, by virtue of planar functions, several infinite families of l-weight linear codes over F p are constructed, where l can be any positive integer and p is a prime number. The weight distributions of these codes are determined completely by utilizing certain approach on exponential sums. Experiments show that some (almost) optimal codes in small dimensions can be produced from our results. Moreover, the related covering codes are also investigated. (c) 2024 Published by Elsevier Inc.
The implicit function theory has many applications in continuous functions as a powerful tool. This paper initiates the research on handling functions over finite fields with characteristic even from an implicit viewp...
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The implicit function theory has many applications in continuous functions as a powerful tool. This paper initiates the research on handling functions over finite fields with characteristic even from an implicit viewpoint, and exploring the applications of implicit functions in cryptographic functions and linear error-correcting codes. The implicit function SG over finite fields is defined by the zeros of a bivariate polynomial G(X,Y). First, we provide basic concepts and constructions of implicit functions. Second, some strong cryptographic functions are constructed by implicit expressions, including semi-bent (or near-bent) balanced Boolean functions and 4differentially uniform involution without fixed points. Moreover, we construct some optimal linear codes and minimal codes by using constructed implicitly defined functions. In our proof, some algebra and algebraic curve techniques over finite fields are used. Finally, some problems for future work are provided. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
linear codes with few weights have many applications in secret sharing, strongly regular graphs, association schemes and authentication codes. Recently, subfield codes of some linear codes with few weights have been s...
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linear codes with few weights have many applications in secret sharing, strongly regular graphs, association schemes and authentication codes. Recently, subfield codes of some linear codes with few weights have been studied. In this paper, we present a family of linear codes with few weights and their punctured codes, whose parameters and weight distributions are completely determined. And we give the parameters of their dual codes, which are length-optimal and dimensional-optimal with respect to the Sphere-packing bound. In addition, we also investigate their subfield codes to obtain some q-ary linear codes with few weights. The weight distributions and dualities of these subfield codes are explicitly determined.
作者：
Li, ShitaoShi, MinjiaLing, SanAnhui Univ
Sch Math Sci Key Lab Intelligent Comp Signal Proc Minist Educ Hefei 230601 Anhui Peoples R China Xidian Univ
State Key Lab Integrated Serv Networks Xian 710071 Peoples R China Nanyang Technol Univ
Sch Phys & Math Sci Singapore 637371 Singapore
The hull of a linear code over a finite field is the intersection of the code and its dual, which was introduced by Assmus and Key to classify finite projective planes. The main objective of this paper is to obtain a ...
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The hull of a linear code over a finite field is the intersection of the code and its dual, which was introduced by Assmus and Key to classify finite projective planes. The main objective of this paper is to obtain a closed mass formula for linear codes with prescribed hull dimension. We simplify the mass formula obtained by Sendrier and provide an alternative proof for the mass formula for self-orthogonal codes obtained by Pless. Finally, we obtain a classification of (optimal) ternary linear codes with small parameters.
In this paper, for n >= 6, we present the generic construction of binary linear codes of length 2(n) - 1 with dimension n + 3, and derive the necessary and sufficient condition for the constructed codes to be minim...
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In this paper, for n >= 6, we present the generic construction of binary linear codes of length 2(n) - 1 with dimension n + 3, and derive the necessary and sufficient condition for the constructed codes to be minimal. Using this generic construction, a new family of minimal binary linear codes violating the Ashikhmin-Barg condition will be constructed from a special class of Boolean functions. We also obtain the weight distribution of the constructed minimal binary linear codes. We will achieve minimal codes with the highest dimension, resulting in a better rate of transmission.
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