The minimum volume of subspace trades

Krotov, Denis S.

doi:10.1016/j.disc.2017.08.012

Cited by 7 publications

(4 citation statements)

References 10 publications

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“…Much less is known for q > 2. It is also known that W q;t,k is full-rank over R, see [3]; the minimum number of non-zeros in a non-void R-null design J q (n, k) → R is t i=0 (1 + q i ) if k = t + 1 and is conjectured to be t i=0 (1 + q i ) if t + 2 ≤ k < n − t, see [2] (with the additional restriction on the null design to be {0, 1, −1}-valued, the conjecture was proved in [4]).…”

Section: On Wilson Matricesmentioning

confidence: 99%

On minimal subspace Zp-null designs

Krotov

2020

Preprint

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Let q be a power of a prime p, and let V be an n-dimensional space over the field GF(q). A Z p -valued function C on the set of k-dimensional subspaces of V is called a k-uniform Z p -null design of strength t if for every t-dimensional subspace y of V the sum of C over the k-dimensional superspaces of y equals 0. For q = p = 2 and 0 ≤ t < k < n, we prove that the minimum number of non-zeros of a non-void k-uniform Z p -null design of strength t equals 2 t+1 . For q > 2, we give lower and upper bounds for that number.

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Section: On Wilson Matricesmentioning

confidence: 99%

On minimal subspace Zp-null designs

Krotov

2020

Preprint

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“…for all t-subspaces T of V . In this situation, the pair (B 1 , B 2 ) has also been called trade or bitrade, see for example [20,21] for some recent results. Furthermore, given integers…”

Section: Preliminariesmentioning

confidence: 99%

A new series of large sets of subspace designs over the binary field

Kiermaier

Laue

Wassermann

2017

Des. Codes Cryptogr.

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In this article, we show the existence of large sets LS 2 [3](2, k, v) for infinitely many values of k and v. The exact condition is v ≥ 8 and 0 ≤ k ≤ v such that for the remaindersv andk of v and k modulo 6 we have 2 ≤v < k ≤ 5.The proof is constructive and consists of two parts. First, we give a computer construction for an LS 2 [3](2, 4, 8), which is a partition of the set of all 4-dimensional subspaces of an 8-dimensional vector space over the binary field into three disjoint 2-(8, 4, 217) 2 subspace designs. Together with the already known LS 2 [3] (2,3,8), the application of a recursion method based on a decomposition of the Graßmannian into joins yields a construction for the claimed large sets.

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“…One of important problems in the theory of bitrades is the problem of finding the minimum sizes of bitrades. This problem was investigated for null designs [4,5,9], for combinatorial bitrades [13], for Latin bitrades [21] and for q-ary Steiner bitrades [17,18]. In this work we study 1-perfect bitrades in the Hamming graph.…”

Section: Introductionmentioning

confidence: 99%

Eigenfunctions and minimum 1-perfect bitrades in the Hamming graph

Valyuzhenich¹

2020

Preprint

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The Hamming graph H(n, q) is the graph whose vertices are the words of length n over the alphabet {0, 1, . . . , q − 1}, where two vertices are adjacent if they differ in exactly one coordinate. The adjacency matrix of H(n, q) has n + 1 distinct eigenvalues n(q − 1) − q • i with corresponding eigenspaces U i (n, q) for 0 ≤ i ≤ n. In this work we study functions belonging to a direct sum U i (n, q) ⊕ U i+1 (n, q) ⊕ . . . ⊕ U j (n, q) for 0 ≤ i ≤ j ≤ n. We find the minimum cardinality of the support of such functions for q = 2 and for q = 3, i + j > n.In particular, we find the minimum cardinality of the support of eigenfunctions from the eigenspace U i (n, 3) for i > n 2 . Using the correspondence between 1-perfect bitrades and eigenfunctions with eigenvalue −1, we find the minimum size of a 1-perfect bitrade in the Hamming graph H(n, 3).

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The minimum volume of subspace trades

Cited by 7 publications

References 10 publications

On minimal subspace Zp-null designs

On minimal subspace Zp-null designs

A new series of large sets of subspace designs over the binary field

Eigenfunctions and minimum 1-perfect bitrades in the Hamming graph

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