An algorithm for computing the weight distribution of a linear [n, k] code over a finite field F-q is developed. The codes are represented by their characteristic vector with respect to a given generator matrix and a ...
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An algorithm for computing the weight distribution of a linear [n, k] code over a finite field F-q is developed. The codes are represented by their characteristic vector with respect to a given generator matrix and a generator matrix of the k-dimensional simplex code S-q,S-k.
In this paper we will analyze one linear code from the theoretical point of view. Namely, the code definition is based on linear quasigroups. In the previous work we classified the quasigroups of order 4 according to ...
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In this paper we will analyze one linear code from the theoretical point of view. Namely, the code definition is based on linear quasigroups. In the previous work we classified the quasigroups of order 4 according to their probability of undetected errors. Now, in this paper we will conclude whether the linear quasigroups that are in the same class in this classification obtain equal number of surely detected incorrectly transmitted bits. Also, we will classify the linear quasigroups of order 4 according to the number of errors that the code surely detects when they are used for coding. At the end we will make conclusion which quasigroups of order 4 are overall best for coding having in mind both important parameters for every code for error detection: the number of errors that the code surely detects and the probability of undetected errors.
Systems glycobiology aims to provide models and analysis tools that account for the biosynthesis, regulation, and interactions with glycoconjugates. To facilitate these methods, there is a need for a clear glycan repr...
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Systems glycobiology aims to provide models and analysis tools that account for the biosynthesis, regulation, and interactions with glycoconjugates. To facilitate these methods, there is a need for a clear glycan representation accessible to both computers and humans. linear code, a linearized and readily parsable glycan structure representation, is such a language. For this reason, linear code was adapted to represent reaction rules, but the syntax has drifted from its original description to accommodate new and originally unforeseen challenges. Here, we delineate the consensuses and inconsistencies that have arisen through this adaptation. We recommend options for a consensus-based extension of linear code that can be used for reaction rule specification going forward. Through this extension and specification of linear code to reaction rules, we aim to minimize inconsistent symbology thereby making glycan database queries easier. With a clear guide for generating reaction rule descriptions, glycan synthesis models will be more interoperable and reproducible thereby moving glycoinformatics closer to compliance with FAIR standards. Here, we present linear code for Reaction Rules (LiCoRR), version 1.0, an unambiguous representation for describing glycosylation reactions in both literature and code.
In this paper we consider an error-detecting code based on linear quasigroups. Namely, each input block a0a1 horizontal ellipsis an-1 is extended into a block a0a1 horizontal ellipsis an-1d0d1 horizontal ellipsis dn-1...
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In this paper we consider an error-detecting code based on linear quasigroups. Namely, each input block a0a1 horizontal ellipsis an-1 is extended into a block a0a1 horizontal ellipsis an-1d0d1 horizontal ellipsis dn-1, where the redundant characters d0,d1, horizontal ellipsis ,dn-1 are defined with di=ai*ai+1*ai+2, where * is a linear quasigroup operation and the operations in the indexes are modulo n. We give a proof that under some conditions the code is linear. Using this fact, we contribute to the determination of the error-detecting capability of the code. Namely, we determine the Hamming distance of the code and from there we obtain the number of errors that the code will detect for sure when linear quasigroups of order 4 from the best class of quasigroups of order 4 for which the constant term in the linear representation is zero matrix are used for coding. All results in the paper are derived for arbitrary length of the input blocks. With the obtained results we showed that when a small linear quasigroup of order 4 from the best class of quasigroups of order 4 is used for coding, the number of errors that the code surely detects is upper bounded with 4.
Reduction of redundancy and improvement of error-correcting capability are essential research themes in the coding theory. The best known codes constructed in various ways are recorded in a database maintained by Mark...
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Reduction of redundancy and improvement of error-correcting capability are essential research themes in the coding theory. The best known codes constructed in various ways are recorded in a database maintained by Markus Grassl. In this paper, we propose an algorithm to construct the best code using punctured codes and a supporting method for constructing the best codes. First, we define a new evaluation function to determine deletion bits and propose an algorithm for constructing punctured linear codes. 27 new best codes were constructed in the proposed algorithm, and 112 new best codes were constructed by further modifying those best codes. Secondly, we evaluate the possibility of increasing the minimum distance based on the relationship between code length, information length, and minimum distance. We narrowed down the target (n, k) code to try the best code search based on the evaluation and found 28 new best codes. We also propose a method to rapidly derive the minimum weight of the modified cyclic codes. A cyclic code loses its cyclic structure when it is modified, so we extend the k-sparse algorithm to use it for modified cyclic codes as well. The extended k-sparse algorithm is used to verify our newly constructed best code.
Monotone span program(MSP) and linear code(LC) are efficient tools to construct linear secret sharing scheme(LSSS) for a given access structure. Since the size of an MSP or the length of an LC corresponds to the commu...
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Monotone span program(MSP) and linear code(LC) are efficient tools to construct linear secret sharing scheme(LSSS) for a given access structure. Since the size of an MSP or the length of an LC corresponds to the communicational complexity of an LSSS, one main motivation to study MSPs or LCs is the lower bound for their sizes or lengths. Therefore, it is one of the most important open problems how to efficiently construct an MSP or LC for a given access structure Γ with the smallest sizes or shortest length. Our contributions are: We extend TANG et al.’s result, showing that, for any given access structureΓ, there exists an MSP or an LC to realize Γ if and only if a system of quadratic equations has solutions; We utilize the relationship between LCs and MSPs to obtain the greatest lower bound on the row size and the column size of MSPs realizing a given Γ, as well as an upper bound on the column size of MSPs; We give an algorithm to construct an MSP with the smallest sizes.
An invariant subcode of a linear block code under the permutation is introduced. The concept of invariant subcode has two types of applications. The first type is decoding of linear block codes given the group of symm...
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ISBN:
(数字)9783319393452
ISBN:
(纸本)9783319393452;9783319393445
An invariant subcode of a linear block code under the permutation is introduced. The concept of invariant subcode has two types of applications. The first type is decoding of linear block codes given the group of symmetry. The second type is the attack the McEliece cryptosystem based on codes correcting errors. Several examples illustrating the concept are presented.
In this article we mainly study linear codes over F(2)n and their binary subfield codes. We construct linear codes over F(2)n whose defining sets are the certain subsets of F(2)(m)n obtained from mathematical objects ...
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In this article we mainly study linear codes over F(2)n and their binary subfield codes. We construct linear codes over F(2)n whose defining sets are the certain subsets of F(2)(m)n obtained from mathematical objects called simplicial complexes. We use a result in LFSR sequences to illustrate the relation of the weights of codewords in two special codes obtained from simplical complexes and then determin the parameters of these codes. We construct fiveinfinite families of distance optimal codes and give sufficient conditions for these codes to be minimal.
We introduce a new class of linear codes over a finite field associated to determinantal varieties in the space of hermitian matrices and determine their length, dimension and minimum distance along with the weight sp...
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We introduce a new class of linear codes over a finite field associated to determinantal varieties in the space of hermitian matrices and determine their length, dimension and minimum distance along with the weight spectrum.
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.
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