Regular two-graphs on 36 vertices

Spence, Edward

doi:10.1016/0024-3795(95)00158-n

Cited by 32 publications

(38 citation statements)

References 2 publications

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“…The equivalence classes of Hadamard matrices of order ≤ 28 have been obtained by Hall [7], [8], Ito et al [10], Kimura [11], [12], [13] and Spence [20]. On the equivalence class of Hadamard matrices, we have the following known results (see [2,Theorem 24.34], [21], [6] 16,20,24,28,32, and 36 are 5, 3, 60, 487, ≥ 66000, and ≥ 200, respectively.…”

Section: Introductionmentioning

confidence: 88%

A sensitive algorithm for detecting the inequivalence of Hadamard matrices

Fang

2003

Math. Comp.

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Abstract. A Hadamard matrix of side n is an n × n matrix with every entry either 1 or −1, which satisfies HH T = nI. Two Hadamard matrices are called equivalent if one can be obtained from the other by some sequence of row and column permutations and negations. To identify the equivalence of two Hadamard matrices by a complete search is known to be an NP hard problem when n increases. In this paper, a new algorithm for detecting inequivalence of two Hadamard matrices is proposed, which is more sensitive than those known in the literature and which has a close relation with several measures of uniformity. As an application, we apply the new algorithm to verify the inequivalence of the known 60 inequivalent Hadamard matrices of order 24; furthermore, we show that there are at least 382 pairwise inequivalent Hadamard matrices of order 36. The latter is a new discovery.

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Section: Introductionmentioning

confidence: 88%

A sensitive algorithm for detecting the inequivalence of Hadamard matrices

Fang

2003

Math. Comp.

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“…Using the obervations made in Section 8.2, the above mentioned symmetric Hadamard matrices of order 36 with constant diagonal that were constructed in [102] lead (by the observations made in Section 8.2) to 227 pairwise non-isomorphic regular (36, 2, 16)-covers. Moreover, the infinite series of symmetric Bush-type Hadamard matrices of order 4n 4 for odd n that was constructed in [82], gives rise to an infinite series of GHM(Z 2 , 4n 4 , 2, 2(n 4 − 1)) and thus to an infinite series of regular (4n 4 , 2, 2(n 4 − 1))-covers (with δ = 2) for all odd n.…”

Section: Summary Of New Resultsmentioning

confidence: 99%

“…Studying the literature about Hadamard matrices, we found that there exist many symmetric Hadamard matrices of order 36 with constant diagonal (cf. [102]). Without loss of generality we can assume that the diagonal is filled with the neutral element of Z 2 (i.e.…”

Section: Summary Of New Resultsmentioning

confidence: 99%

A new construction of antipodal distance regular covers of complete graphs through the use of Godsil-Hensel matrices

Klin

Pech²

2011

Ars Math. Contemp.

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New constructions of regular distance regular antipodal covers (in the sense of GodsilHensel) of complete graphs K n are presented. The main source of these constructions are skew generalized Hadamard matrices. It is described how to produce such a matrix of order n 2 over a group T from an arbitrary given generalized Hadamard matrix of order n over the same group T . Further, a new regular cover of K 45 on 135 vertices is produced with the aid of a decoration of the alternating group A 6 .

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“…The first steps of the proof parallel the one for the preceding proposition, the only difference being that after switching to obtain the isolated vertex v 1 , the valency of the other vertices is For n = 36, k = 21, the argument in Proposition 25 does not guarantee the existence of induced complete bipartite subgraphs on 5 vertices. However, by having a computer search all 227 known equivalence classes [17], one finds that all members have at least one induced complete bipartite subgraph on 6 vertices. Thus, the m-deletion error is the same for all 2-uniform (36, 21)-frames, If n = 36, k = 15, then Proposition 26 shows that each graph G F contains an induced complete bipartite on 5 vertices.…”

Section: Definition 14mentioning

confidence: 99%

Loss-insensitive vector encoding with two-uniform frames

Bodmann

Paulsen

2005

Wavelets XI

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Abstract. The central topic of this paper is the linear, redundant encoding of vectors using frames for the purpose of loss-insensitive data transmission. Our goal is to minimize the reconstruction error when frame coefficients are accidentally erased. Two-uniform frames are known to be optimal for handling up to two erasures, in the sense that they minimize the largest Euclidean error norm when up to two frame coefficients are set to zero. Here, we consider the case when an arbitrary number of the frame coefficients of a vector is lost. We derive general error bounds and apply these to concrete examples. We show that among the 227 known equivalence classes of two-uniform (36,15)-frames arising from Hadamard matrices, there are 5 that give smallest error bounds for up to 8 erasures.

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Regular two-graphs on 36 vertices

Cited by 32 publications

References 2 publications

A sensitive algorithm for detecting the inequivalence of Hadamard matrices

A sensitive algorithm for detecting the inequivalence of Hadamard matrices

A new construction of antipodal distance regular covers of complete graphs through the use of Godsil-Hensel matrices

Loss-insensitive vector encoding with two-uniform frames

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