Sudoku-like arrays, codes and orthogonality

Huggan, Melissa A.; Mullen, Gary L.; Stevens, Brett; Thomson, David J.

doi:10.1007/s10623-016-0190-y

Cited by 10 publications

(5 citation statements)

References 12 publications

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“…Partitions of the Hamming space into MDS codes (including the ones considered in Section 4.2) are a special subcase of more general family of Sudoku-like arrays considered in [10]. The question which of such partitions can be lengthened is of independent interest, while finding non-lengthenable partitions into (q, q q−2 , 3) codes, q > 4 would result in non-lengthenable codes with parameters of shortened 1-perfect codes by Theorem 3.…”

Section: Discussionmentioning

confidence: 99%

On $q$-ary shortened-$1$-perfect-like codes

Shi,

Wu,

Krotov

2021

Preprint

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We study codes with parameters of q-ary shortened Hamming codes, i.e., (n = (q m − q)/(q − 1), q n−m , 3) q . At first, we prove the fact mentioned in [A. E. Brouwer et al. Bounds on mixed binary/ternary codes. IEEE Trans. Inf. Theory 44 (1998) 140-161] that such codes are optimal, generalizing it to a bound for multifold packings of radius-1 balls, with a corollary for multiple coverings. In particular, we show that the punctured Hamming code is an optimal q-fold packing with minimum distance 2. At second, we show the existence of 4-ary codes with parameters of shortened 1-perfect codes that cannot be obtained by shortening a 1-perfect code.

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Section: Discussionmentioning

confidence: 99%

On $q$-ary shortened-$1$-perfect-like codes

Shi,

Wu,

Krotov

2021

Preprint

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“…The existence of (s, f )-hypercubes for given parameters is an open problem, and there is no (s, f )-latin hypercube of order n and dimension d for s ≥ 2 and f > (n − 1) s−1 (See [7,Lemma 2.2]. For further results on latin cubes, see [1,4,6,11,15] and references therein.…”

Section: Introductionmentioning

confidence: 99%

Symmetric Layer-Rainbow Colorations of Cubes

Bahmanian¹

2022

Preprint

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Can we color the n 3 cells of an n × n × n cube L with n 2 colors in such a way that each layer parallel to each face contains each color exactly once and that the coloring is symmetric so that L ij = L j i = L ij for distinct i, j, ∈ {1, . . . , n}, and L iij = L jji , L iji = L jij , L ijj = L jii for i, j ∈ {1, . . . , n}? Using transportation networks, we show that such a coloring is possible if and only if n ≡ 0, 2 (mod 3) (with two exceptions, n = 1 and n = 3). Motivated by the designs of experiments, the study of these objects (without symmetry) was initiated by Kishen and Fisher in the 1940's. These objects are also closely related to orthogonal arrays whose existence has been extensively investigated, and they are natural three-dimensional analogues of symmetric latin squares.

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“…Sudoku tables are a special class of Latin squares which are very popular among researchers [3], [8], [9], [13], [14]. A Sudoku table of order m with subtables of order s × d is a Latin square in which m different numbers occur exactly once in each row, column, and subtable [6].…”

Section: Introductionmentioning

confidence: 99%

Construction of Standard Solid Sudoku Cubes and 3D Sudoku Puzzles

2022

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In this paper, standard solid Sudoku cubes (SSSCs), a three-dimensional (3D) extension of Sudoku tables, are introduced, and a method to construct these cubes is presented. This is the first class of standard solid Sudoku cubes. An SSSC of order m is a solid Latin cube of order m with solid subcubes of order x × y × z in which each element occurs exactly once in each row, column, depth, and subcube. The structure of these cubes is based on cyclotomic cosets of Z n , and we make use of a design vector Z and a basic table T to construct SSSCs. We obtain m tables by multiplying all entries of T with a number from the vector Z. Then, these tables are converted to an SSSC by stacking them in order. Based on this method of construction, a perfect set of strongly mutually distinct standard solid Sudoku cubes is designed. We also provide a two-dimensional (2D) representation of these SSSCs in a table with numbers placed on the main diagonal of its subtables. Finally, a new class of 3D Sudoku puzzles based on the SSSC is presented as standard solid Sudoku puzzles (SSSPs).

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Sudoku-like arrays, codes and orthogonality

Cited by 10 publications

References 12 publications

On $q$-ary shortened-$1$-perfect-like codes

On $q$-ary shortened-$1$-perfect-like codes

Symmetric Layer-Rainbow Colorations of Cubes

Construction of Standard Solid Sudoku Cubes and 3D Sudoku Puzzles

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