On the structure of cube tiling codes

Kisielewicz, Andrzej

doi:10.1016/j.ejc.2020.103168

Cited by 4 publications

(5 citation statements)

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“…In [20] Mathew, Östergård and Popa gave an affirmative answer for d = 5 (they called gluing and cutting shifting). In [12] we proved that the above results are true for any S .…”

Section: Introductionmentioning

confidence: 53%

“…By Theorem 1, there is precisely one, up to isomorphism, such pair of codes, where |V | ≤ 15. But none of the codes from the family 12 v has a form of one of the codes enumerated in Theorem 1.…”

Section: Outline Of the Proof Of Theorem 3 And Results Of The Computa...mentioning

confidence: 99%

“…In particular, for cube tiling codes and dimensions d ≥ 4 it is rather worthless. For example, if V ⊂ S 4 is a cube tiling code and |S | = 16, then m > 10 16 ( [12]), and if V ⊂ S 5 is a cube tiling code and S = {a , a ′ , b , b ′ }, then m > 6 • 10 14 ( [20]). (Let us recall that every cube tilings codes V , W ⊂ S d are strongly equivalent for d ≤ 5 ([5, 12, 20]), and therefore m is the number of all cube tiling codes.…”

Section: Algorithm Glueandcutmentioning

confidence: 99%

“…A polybox code V ⊂ S d is a cube tiling code if |V | = 2 d . (It is easy to see that a cube tiling code V ⊂ S d induces an r -perfect code C ⊂ d 4r +2 , r ∈ , in the maximum metric ( [3,12])). A natural interpretation of a polybox code is some system of boxes.…”

Section: Introductionmentioning

confidence: 99%

“…REMARK 1 We emphasize that passing from a twin pair v, w with v i = w ′ i to a twin pair q , t with q i = t ′ i , q i ∈ {v i , v ′ i } and v j = q j for all j = i , we may create the gluing u, because sometimes application of the gluing operation (without cutting) may lead to an interesting observation on the structure of polybox codes, as it was done in [12]. However, in the presented paper we shall not transform proper polybox codes into improper (that is codes containing words u ∈ ( * S )…”

Section: Introductionmentioning

confidence: 99%

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Gluing and cutting cube tiling codes in dimension six

Kisielewicz¹

2020

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Let S be a set of arbitrary objects, and let s → s ′ be a permutation of S such that s ′′ = (s ′ ) ′ = s and sand they form a twin pair if v ′ i = w i and v j = w j for every j ∈periodic cube tiling of d and a cube tiling of flat torus d can be encoded in a form of a cube tiling code. A twin pair v, w in which v i = w ′ i is glue (at the i th position) if the pair v, w is replaced by one word u such that u j = v j = w j for every j ∈ [d ] \ {i } and u i = * , where * ∈ S is some extra fixed symbol. A word u with u i = * is cut (at the i th position) if u is replaced by a twin pair q , t such that q i = t ′ i and u j = q j = t j for every j ∈ [d ] \ {i }. If V , W ⊂ S d are two cube tiling codes and there is a sequence of twin pairs which can be interchangeably gluing and cutting in a way which allows us to pass from V to W , then we say that W is obtained from V by gluing and cutting. In the paper it is shown that for every two cube tiling codes in dimension six one can be obtained from the other by gluing and cutting.

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“…In [20] Mathew, Östergård and Popa gave an affirmative answer for d = 5 (they called gluing and cutting shifting). In [12] we proved that the above results are true for any S .…”

Section: Introductionmentioning

confidence: 53%

Section: Outline Of the Proof Of Theorem 3 And Results Of The Computa...mentioning

confidence: 99%