scholarly journals Hadamard matrices of order 36 and double-even self-dual [72,36,12] codes

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Iliya Bouyukliev ◽  
Veerle Fack ◽  
Joost Winne
Keyword(s):  
Fixed Points ◽  
Hadamard Matrices ◽  
Dual Codes ◽  
International Audience

International audience Before this work, at least 762 inequivalent Hadamard matrices of order 36 were known. We found 7238 Hadamard matrices of order 36 and 522 inequivalent [72,36,12] double-even self-dual codes which are obtained from all 2-(35,17,8) designs with an automorphism of order 3 and 2 fixed points and blocks.

scholarly journals Construction of new self-dual codes over $GF(5)$ using skew-Hadamard matrices

2009 ◽  
Vol 3 (3) ◽  
pp. 251-263 ◽  
Author(s):  
Christos Koukouvinos ◽  
◽  
Dimitris E. Simos
Keyword(s):  
Hadamard Matrices ◽  
Dual Codes

Extremal doubly-even self-dual codes from Hadamard matrices

2006 ◽  
Vol 9 (2) ◽  
pp. 331-339
Author(s):  
S. Georgiou ◽  
C. Koukouvinos ◽  
E. Lappas
Keyword(s):  
Hadamard Matrices ◽  
Dual Codes

scholarly journals A new generation tree for permutations, preserving the number of fixed points

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Philippe Duchon ◽  
Romaric Duvignau
Keyword(s):  
Fixed Points ◽  
Random Number ◽  
Random Number Generator ◽  
Specific Property ◽  
Number Generator ◽  
Random Generation ◽  
Combinatorial Algorithm ◽  
International Audience ◽  
Uniform Generation ◽  
New Generation

International audience We describe a new uniform generation tree for permutations with the specific property that, for most permutations, all of their descendants in the generation tree have the same number of fixed points. Our tree is optimal for the number of permutations having this property. We then use this tree to describe a new random generation algorithm for derangements, using an expected n+O(1) calls to a random number generator. Another application is a combinatorial algorithm for exact sampling from the Poisson distribution with parameter 1.

scholarly journals The $q=-1$ phenomenon for bounded (plane) partitions via homology concentration

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
P. Hersh ◽  
J. Shareshian ◽  
D. Stanton
Keyword(s):  
Fixed Points ◽  
General Framework ◽  
Plane Partitions ◽  
International Audience

International audience Algebraic complexes whose "faces'' are indexed by partitions and plane partitions are introduced, and their homology is proven to be concentrated in even dimensions with homology basis indexed by fixed points of an involution, thereby explaining topologically two quite important instances of Stembridge's $q=-1$ phenomenon. A more general framework of invariant and coinvariant complexes with coefficients taken $\mod 2$ is developed, and as a part of this story an analogous topological result for necklaces is conjectured. Complexes algébriques dont les "faces'' sont indexées par des partitions et des partitions planes sont introduits. Il est démontré que leur homologie est concentrée en dimensions paires, avec base de homologie indexée par des points fixes d'une involution. Ce résultat explique d'une manière topologique deux instances du phénomène $q=-1$ dû a Stembridge. De plus, un cadre plus général des complexes invariants et coinvariants dont les coefficients sont pris modulo $2$ est développé. Comme part de cette histoire, nous conjecturons un résultat analogue pour des colliers.

scholarly journals Latin square Thue-Morse sequences are overlap-free

2007 ◽  
Vol Vol. 9 no. 1 (Automata, Logic and Semantics) ◽  
Author(s):  
C. Robinson Tompkins
Keyword(s):  
Fixed Points ◽  
Latin Square ◽  
International Audience

Automata, Logic and Semantics International audience We define a morphism based upon a Latin square that generalizes the Thue-Morse morphism. We prove that fixed points of this morphism are overlap-free sequences, generalizing results of Allouche - Shallit and Frid.

scholarly journals Cyclic Sieving, Promotion, and Representation Theory

2008 ◽  
Vol DMTCS Proceedings vol. AJ,... (Proceedings) ◽  
Author(s):  
Brendon Rhoades
Keyword(s):  
Representation Theory ◽  
Fixed Points ◽  
Canonical Basis ◽  
Dihedral Groups ◽  
Jeu De Taquin ◽  
Noncrossing Partitions ◽  
Dual Canonical Basis ◽  
International Audience ◽  
Cyclic Sieving

International audience We prove a collection of conjectures due to Abuzzahab-Korson-Li-Meyer, Reiner, and White regarding the cyclic sieving phenomenon as it applies to jeu-de-taquin promotion on rectangular tableaux. To do this, we use Kazhdan-Lusztig theory and a characterization of the dual canonical basis of $\mathbb{C}[x_{11}, \ldots , x_{nn}]$ due to Skandera. Afterwards, we extend our results to analyzing the fixed points of a dihedral action on rectangular tableaux generated by promotion and evacuation, suggesting a possible sieving phenomenon for dihedral groups. Finally, we give applications of this theory to cyclic sieving phenomena involving reduced words for the long elements of hyperoctohedral groups, handshake patterns, and noncrossing partitions.

scholarly journals The Bruhat order on conjugation-invariant sets of involutions in the symmetric group

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Mikael Hansson
Keyword(s):  
Fixed Point ◽  
Symmetric Group ◽  
Fixed Points ◽  
Bruhat Order ◽  
Rank Function ◽  
Invariant Sets ◽  
Graded Poset ◽  
International Audience ◽  
Complete Characterisation

12 pages, 3 figures International audience Let $I_n$ be the set of involutions in the symmetric group $S_n$, and for $A \subseteq \{0,1,\ldots,n\}$, let \[ F_n^A=\{\sigma \in I_n \mid \text{$\sigma$ has $a$ fixed points for some $a \in A$}\}. \] We give a complete characterisation of the sets $A$ for which $F_n^A$, with the order induced by the Bruhat order on $S_n$, is a graded poset. In particular, we prove that $F_n^{\{1\}}$ (i.e., the set of involutions with exactly one fixed point) is graded, which settles a conjecture of Hultman in the affirmative. When $F_n^A$ is graded, we give its rank function. We also give a short new proof of the EL-shellability of $F_n^{\{0\}}$ (i.e., the set of fixed point-free involutions), which was recently proved by Can, Cherniavsky, and Twelbeck. Soit $I_n$ l’ensemble d’involutions dans le groupe symétrique $S_n$, et pour $A \subseteq \{0,1,\ldots,n\}$, soit\[ F_n^A=\{\sigma \in I_n \mid \text{$\sigma$ a $a$ points fixes pour quelque $a \in A$}\}. \] Nous caractérisons tous les ensembles $A$ dont les $F_n^A$ , avec l’ordre induit par l’ordre de Bruhat sur $S_n$, est un posetgradué. En particulier, nous démontrons que $F_n^{\{1\}}$ (c’est-à-dire, l’ensemble d’involutions avec précis en point fixe)est gradué, ce qui résout une conjecture d’Hultman à l’affirmative. Lorsque $F_n^A$ est gradué, nous donnons sa fonctionde rang. En plus, nous donnons une nouvelle démonstration courte l’EL-shellability de $F_n^{\{0\}}$ (c’est-à-dire, l’ensembled’involutions sans points fixes), établie récemment par Can, Cherniavsky et Twelbeck.

scholarly journals Cyclic Sieving and Plethysm Coefficients

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
David B Rush
Keyword(s):  
Fixed Points ◽  
Schur Function ◽  
Young Tableaux ◽  
Jeu De Taquin ◽  
Cyclic Action ◽  
Domino Tableaux ◽  
Ribbon Tableaux ◽  
International Audience ◽  
Cyclic Sieving ◽  
Element Set

International audience A combinatorial expression for the coefficient of the Schur function $s_{\lambda}$ in the expansion of the plethysm $p_{n/d}^d \circ s_{\mu}$ is given for all $d$ dividing $n$ for the cases in which $n=2$ or $\lambda$ is rectangular. In these cases, the coefficient $\langle p_{n/d}^d \circ s_{\mu}, s_{\lambda} \rangle$ is shown to count, up to sign, the number of fixed points of an $\langle s_{\mu}^n, s_{\lambda} \rangle$-element set under the $d^e$ power of an order $n$ cyclic action. If $n=2$, the action is the Schützenberger involution on semistandard Young tableaux (also known as evacuation), and, if $\lambda$ is rectangular, the action is a certain power of Schützenberger and Shimozono's <i>jeu-de-taquin</i> promotion.This work extends results of Stembridge and Rhoades linking fixed points of the Schützenberger actions to ribbon tableaux enumeration. The conclusion for the case $n=2$ is equivalent to the domino tableaux rule of Carré and Leclerc for discriminating between the symmetric and antisymmetric parts of the square of a Schur function. Une expression combinatoire pour le coefficient de la fonction de Schur $s_{\lambda}$ dans l’expansion du pléthysme $p_{n/d}^d \circ s_{\mu}$ est donné pour tous $d$ que disent $n$, dans les cas où $n=2$, ou $\lambda$ est rectangulaire. Dans ces cas, le coefficient $\langle p_{n/d}^d \circ s_{\mu}, s_{\lambda} \rangle$ se montre à compter, où l’on ignore le signe, le nombre des point fixés d’un ensemble de $\langle s_{\mu}^n, s_{\lambda} \rangle$ éléments sous la puissance $d^e$ d’une action cyclique de l’ordre $n$. Si $n=2$, l’action est l’involution de Schützenberger sur les tableaux semi-standard de Young (aussi connu sous le nom des évacuations), et si $\lambda$ est rectangulaire, l’action est une certaine puissance de l’avancement jeu-de-taquin de Schützenberger et Shimozono.Ce travail étend les résultats de Stembridge et Rhoades, liant les point fixés des actions de Schützenberger aux tableaux de ruban. Pour le cas $n=2$ , la conclusion est équivalent à la règle des tableaux de dominos de Carré et Leclerc, qui distingue entre les parties symétriques et asymétriques du carré d’une fonction de Schur.

scholarly journals Classification of Generalized Hadamard Matrices $H(6,3)$ and Quaternary Hermitian Self-Dual Codes of Length 18

10.37236/443 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Masaaki Harada ◽  
Clement Lam ◽  
Akihiro Munemasa ◽  
Vladimir D. Tonchev
Keyword(s):  
Hadamard Matrices ◽  
Dual Codes ◽  
Transversal Designs

All generalized Hadamard matrices of order 18 over a group of order 3, $H(6,3)$, are enumerated in two different ways: once, as class regular symmetric $(6,3)$-nets, or symmetric transversal designs on 54 points and 54 blocks with a group of order 3 acting semi-regularly on points and blocks, and secondly, as collections of full weight vectors in quaternary Hermitian self-dual codes of length 18. The second enumeration is based on the classification of Hermitian self-dual $[18,9]$ codes over $GF(4)$, completed in this paper. It is shown that up to monomial equivalence, there are 85 generalized Hadamard matrices $H(6,3)$, and 245 inequivalent Hermitian self-dual codes of length 18 over $GF(4)$.

Sign in / Sign up

Export Citation Format

Share Document