Binary Perfect Codes of Length 15 by the Generalized Concatenated Construction

Zinoviev, V. A.; Zinoviev, Dmitrii

doi:10.1023/b:prit.0000024877.03232.39

Cited by 36 publications

(21 citation statements)

References 7 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…(1) Add two codewords, (1111 | 0000) and (0000 | 1111), to C. It is clear that the resulting code is a constant-weight (8,4,4,14) code, which is equivalent to S(8, 4, 3) [15]. Now the result follows from the uniqueness of S(8, 4, 3) [4].…”

Section: Systems Sqs(v) Obtained By the Gc-constructionmentioning

confidence: 93%

“…Then there are i, j, 1 ≤ i < j ≤ 4, for which W i,j has at least 4 codewords; i.e., we get a contradiction to Proposition 4. Proposition 6 [14]. Let W be a (4, 2, 2, N w ) 4 constant-weight code.…”

Section: Proofmentioning

confidence: 98%

“…It is known (see [14,Section 1]) that there are 5 nonequivalent MDS (4, 2, 64) 4 codes A i over the alphabet E a . They are given in [14] in the so-called canonical forms.…”

Section: Mds (4 2 64) 4 Codesmentioning

confidence: 99%

“…This construction comes from the GC-construction of binary nonlinear extended perfect codes [14,16], i.e., binary codes A (16, 4, 2 11 ). It is well known that codewords of weight 4 of such a code C form a constant-weight code C (16,4,4,140), i.e., a Steiner system S (16,4,3).…”

Section: Systems Sqs(v) Obtained By the Gc-constructionmentioning

confidence: 99%

See 3 more Smart Citations

Classification of steiner quadruple systems of order 16 and rank at most 13

Zinoviev

2004

Probl Inf Transm

Self Cite

View full text Add to dashboard Cite

A Steiner quadruple system SQS(v) of order v is a 3-design T (v, 4, 3, λ) with λ = 1. In this paper we describe all nonisomorphic systems SQS(16) that can be obtained by the generalized concatenated construction (GC-construction). These Steiner systems have rank at most 13 over F 2 . In particular, there is one system SQS(16) of rank 11 (points and planes of the affine geometry AG(4, 2)), fifteen systems of rank 12, and 4131 systems of rank 13. All these Steiner systems are resolvable.

show abstract

Section: Systems Sqs(v) Obtained By the Gc-constructionmentioning

confidence: 93%

Section: Proofmentioning

confidence: 98%

“…It is known (see [14,Section 1]) that there are 5 nonequivalent MDS (4, 2, 64) 4 codes A i over the alphabet E a . They are given in [14] in the so-called canonical forms.…”

Section: Mds (4 2 64) 4 Codesmentioning

confidence: 99%

Section: Systems Sqs(v) Obtained By the Gc-constructionmentioning

confidence: 99%

See 2 more Smart Citations

Classification of steiner quadruple systems of order 16 and rank at most 13

Zinoviev

2004

Probl Inf Transm

Self Cite

View full text Add to dashboard Cite

show abstract

“…They did not establish the number of isotopism classes for order 5, but found the number for orders 1 to 4 respectively to be 1, 1, 1, 12. Zinoviev and Zinoviev [28] found that there are 5 paratopy classes of latin cubes of order 4. Our computations confirm the above numbers.…”

Section: Basic Definitionsmentioning

confidence: 99%

A Census of Small Latin Hypercubes

McKay¹,

Wanless²

2008

SIAM J. Discrete Math.

View full text Add to dashboard Cite

We count all latin cubes of order n ≤ 6 and latin hypercubes of order n ≤ 5 and dimension d ≤ 5. We classify these (hyper)cubes into isotopy classes and paratopy classes (main classes). For the same values of n and d we classify all d-ary quasigroups of order n into isomorphism classes and also count them according to the number of identity elements they possess (meaning we have counted the d-ary loops).We also give an exact formula for the number of (isomorphism classes of) d-ary quasigroups of order 3 for every d. Then we give a number of constructions for d-ary quasigroups with a specific number of identity elements. In the process, we prove that no 3-ary loop of order n can have exactly n − 1 identity elements (but no such result holds in dimensions other than 3). Finally, we give some new examples of latin cuboids which cannot be extended to latin cubes.

show abstract