On enumeration of the perfect binary codes of length 15

Malyugin, S. A.

doi:10.1016/s0166-218x(02)00303-7

Cited by 11 publications

(9 citation statements)

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“…That means it does not belong to the set of all perfect codes obtained by switchings from the Hamming code. Malyugin [47] enumerated all perfect codes of length 15 obtained from the Hamming code by simultaneous switchings of nonintersecting components using different coordinates. The number of such different codes is 131 224 432.…”

Section: Theorem 3 (See Krotovmentioning

confidence: 99%

“…For n = 15 full rank perfect codes with kernel dimension 1 k 5 are known, see [28,30,54], for k 6 such codes do not exist [55]. For r < 15 perfect codes of length 15 for all possible pairs (r, k) are given in [51,59].…”

Section: Theorem 8 (See Phelps and Villanuevamentioning

confidence: 99%

“…Malyugin [51] proved that there are 12 inequivalent Vasil'ev codes of length 15 with rank 12, see also [102].…”

Section: Theorem 8 (See Phelps and Villanuevamentioning

confidence: 99%

“…As a consequence upper bounds on the order of the automorphism groups of an extended perfect binary code, see [49,81,82], and a Steiner system S(t, t + 1, n), see [81], are derived. There are some other new results concerning automorphism groups of perfect codes, see [20,21,52,76]. In [78] applying some well-known constructions (Vasil'ev's, Plotkin's and Mollard's) to known binary transitive codes of some lengths and using some additional conditions infinite classes of transitive binary codes (not necessarily perfect codes) of greater lengths are presented.…”

Section: Automorphism Groups Of Perfect Codesmentioning

confidence: 99%

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On perfect binary codes

Solov’eva

2008

Discrete Applied Mathematics

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Some results on perfect codes obtained during the last 6 years are discussed. The main methods to construct perfect codes such as the method of -components and the concatenation approach and their implementations to solve some important problems are analyzed. The solution of the ranks and kernels problem, the lower and upper bounds of the automorphism group order of a perfect code, spectral properties, diameter perfect codes, isometries of perfect codes and codes close to them by close-packed properties are considered.The topic of perfect codes is one of the most investigated recently topics in coding theory. The class of perfect binary codes is very complicated, large and intensively studied by many researches. The investigation of nontrivial properties of perfect codes is important both from coding point of view (for the solution of the classification problem for such codes) and for combinatorics, graph theory, group theory, geometry.There are several surveys devoted to perfect codes, see [17,25,39,40,[72][73][74]77], so we discuss in this paper only some results on perfect binary codes obtained in the last 6 years. The paper is organized as follows: first we give necessary definitions (Section 1), then consider some methods to construct perfect codes such as the method of -components, the method of i-components and the concatenation approach and their implementations to get solutions of some important problems (see Sections 2, 3 and 4, respectively). Next we discuss the solution of the ranks and kernels problem (Section 5), spectral properties (Section 6), automorphism groups of perfect codes (Section 7), isometries of perfect codes and codes close to them by close-packed properties (Section 8). Some results we discuss in more details (Sections 2-6), others we mention more briefly (Sections 7-9).

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Section: Theorem 3 (See Krotovmentioning

confidence: 99%

Section: Theorem 8 (See Phelps and Villanuevamentioning

confidence: 99%

“…Malyugin [51] proved that there are 12 inequivalent Vasil'ev codes of length 15 with rank 12, see also [102].…”

Section: Theorem 8 (See Phelps and Villanuevamentioning

confidence: 99%

Section: Automorphism Groups Of Perfect Codesmentioning

confidence: 99%

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On perfect binary codes

Solov’eva

2008

Discrete Applied Mathematics

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“…The converse is not true in general. There exist transitive codes such that codes obtained from them by puncturing some coordinate are not transitive; for instance, Malyugin [7] found an extended perfect transitive code of length 16 such that each of its sixteen punctured codes of length 15 is not transitive. Thus, extended transitive codes should be constructed and studied independently of constructing nonextended transitive codes.…”

Section: Constructions Of Transitive Codesmentioning

confidence: 99%