Classification of Graeco-Latin Cubes

Kokkala, Janne I.; Östergård, Patric R. J.

doi:10.1002/jcd.21400

Cited by 11 publications

(13 citation statements)

References 23 publications

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“…, 8, respectively. For n = 5, the codes are equivalent to Graeco-Latin cubes which have been classified recently [13]; there are 1, 1, and 12484 equivalence classes of such codes for q = 5, 7, 8, respectively. This work consists of two parts.…”

Section: Introductionmentioning

confidence: 99%

On the Classification of MDS Codes

Kokkala

Krotov

Östergård

2015

IEEE Trans. Inform. Theory

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A q-ary code of length n, size M , and minimum distance d is called an (n, M, d)q code. An (n, q k , n − k + 1)q code is called a maximum distance separable (MDS) code. In this work, some MDS codes over small alphabets are classified. It is shown that every7} is equivalent to a linear code with the same parameters. This implies that the (6, 5 4 , 3)5 code and the (n, 7 n−2 , 3)7 MDS codes for n ∈ {6, 7, 8} are unique. The classification of one-errorcorrecting 8-ary MDS codes is also finished; there are 14, 8, 4, and 4 equivalence classes of (n, 8 n−2 , 3)8 codes for n = 6, 7, 8, 9, respectively. One of the equivalence classes of perfect (9,8 7 , 3)8 codes corresponds to the Hamming code and the other three are nonlinear codes for which there exists no previously known construction.

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Section: Introductionmentioning

confidence: 99%

On the Classification of MDS Codes

Kokkala

Krotov

Östergård

2015

IEEE Trans. Inform. Theory

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“…In particular, fixing any k 3 arguments of both f i and f j , i ¤ j , always results in a pair of orthogonal latin cubes. By the previous computational results [5], every such pair of order 5 or 7 is linear. By Lemma 12, the .d 1/-tuple of latin hypercubes .f 1 ; : : : ; f d 1 / is linear, which means that the code is equivalent to a linear code.…”

Section: Lemma 11mentioning

confidence: 93%

“…We do this by finding sets of .n 1; q n 3 ; 3/ q codes fC We perform isomorph rejection on the seeds and the full codes. To determine whether two codes are equivalent, we transfer the codes to graphs as in, for example, [5], and use nauty [9] to solve the graph isomorphism problem. The software also produces the automorphism group of a graph, so it can be used to find all automorphisms of a code.…”

Section: Algorithm For Exhaustive Generationmentioning

confidence: 99%

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Classification of MDS Codes over Small Alphabets

Kokkala

Krotov

Östergård

2015

Coding Theory and Applications

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A q-ary code of length n, size M , and minimum distance d is called an .n; M; d / q code. An .n; q k ; d / q code with d D n k C 1 is said to be maximum distance separable (MDS). Here we show that every code with parameters .k C d 1; q k ; d / q where k; d 3 and q D 5; 7, is equivalent to a linear code, which implies that the .6; 5 4 ; 3/ 5 code and the .n; 7 n 2 ; 3/ 7 codes for n D 6; 7; 8 are unique. We also show that there are 14, 8, 4, and 4 equivalence classes of .n; 8 n 2 ; 3/ 8 codes for n D 6; 7; 8; 9, respectively. This work is continuation of a previous article classifying .5; q 3 ; 3/ q codes for q D 5; 7; 8.

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“…They proceed by generating the squares row by row using an algorithms of Sade [29,37]. Niskanen and Östergård's clique finding software, cliquer has been successfully used to find mutually orthogonal latin cubes [21,35]. Kidd [20] and Benadé, Burger, and van Vuuren [3] use custom-written backtracking searches to enumerate all triples of MOLS up to order 8.…”

Section: Introductionmentioning

confidence: 99%

Integer and Constraint Programming Revisited for Mutually Orthogonal Latin Squares

Rubin,

Bright,

Cheung

et al. 2021

Preprint

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In this paper we provide results on using integer programming (IP) and constraint programming (CP) to search for sets of mutually orthogonal latin squares (MOLS). Both programming paradigms have previously successfully been used to search for MOLS, but solvers for IP and CP solvers have significantly improved in recent years and data on how modern IP and CP solvers perform on the MOLS problem is lacking. Using state-of-the-art solvers as black boxes we were able to quickly find pairs of MOLS (or prove their nonexistence) in all orders up to ten. Moreover, we improve the effectiveness of the solvers by formulating an extended symmetry breaking method as well as an improvement to the straightforward CP encoding. We also analyze the effectiveness of using CP and IP solvers to search for triples of MOLS, compare our timings to those which have been previously published, and estimate the running time of using this approach to resolve the longstanding open problem of determining the existence of a triple of MOLS of order ten.

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Classification of Graeco-Latin Cubes

Cited by 11 publications

References 23 publications

On the Classification of MDS Codes

On the Classification of MDS Codes

Classification of MDS Codes over Small Alphabets

Integer and Constraint Programming Revisited for Mutually Orthogonal Latin Squares

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