Translation invariant valuations in the space of planes

The paper is related to the problem of measure generation in the space IE of planes in IR ³ by combinatorial, translation invariant valuations. General results concerning that problem have been derived in 1994 and 1996 (this journal vol. 31, no. 4. and vol. 33, no. 4, respectively). The purpose of the present article is to give a proof of two geometrical identities on which the theorem on valuations in the space IE can be based. The article consists of the motivational first part that contains the basic concepts from the theory of combinatorial valuations and measure generation in IE, and the second that gives a proof of the identities in question.     

Construction of MDS self-dual codes over Galois rings

The purpose of this paper is to construct nontrivial MDS self-dual codes over Galois rings. We consider a building-up construction of self-dual codes over Galois rings as a GF(q)-analogue of (Kim and Lee, J Combin Theory ser A, 105:79–95). We give a necessary and sufficient condition on which the building-up construction holds. We construct MDS self-dual codes of lengths up to 8 over GR(3²,2), GR(3³,2) and GR(3⁴,2), and near-MDS self-dual codes of length 10 over these rings. In a similar manner, over GR(5²,2), GR(5³,2) and GR(7²,2), we construct MDS self-dual codes of lengths up to 10 and near-MDS self-dual codes of length 12. Furthermore, over GR(11²,2) we have MDS self-dual codes of lengths up to 12.      

Improving the extraction and expansion method for large graph coloring

Graph coloring is one of the most studied combinatorial optimization problems. This paper presents an improved extraction and expansion method (IE²COL), initially introduced in Wu and Hao (2012) [44]. IE²COL employs a forward independent set extraction strategy to reduce the initial graph $G$. From the reduced graph, IE²COL triggers a backward coloring process which uses extracted independent sets as new color classes for intermediate subgraph coloring. The proposed method is assessed on 20 large benchmark graphs with 900 to 4000 vertices. Computational results show that it provides new upper bounds for 6 graphs and consistently matches the current best-known results for 12 other graphs.     

Directed graph representation of half-rate additive codes over GF(4)

We show that (n, 2 ⁿ ) additive codes over GF(4) can be represented as directed graphs. This generalizes earlier results on self-dual additive codes over GF(4), which correspond to undirected graphs. Graph representation reduces the complexity of code classification, and enables us to classify additive (n, 2 ⁿ ) codes over GF(4) of length up to 7. From this we also derive classifications of isodual and formally self-dual codes. We introduce new constructions of circulant and bordered circulant directed graph codes, and show that these codes will always be isodual. A computer search of all such codes of length up to 26 reveals that these constructions produce many codes of high minimum distance. In particular, we find new near-extremal formally self-dual codes of length 11 and 13, and isodual codes of length 24, 25, and 26 with better minimum distance than the best known self-dual codes.     

Minimal ellipsoids and their duals

It is proved that “most” convex bodies in IE ^d touch the boundaries of their minimal circumscribed and their maximal inscribed ellipsoids in preciselyd(d+3)/2 points. A version of the former result shows that for “most” compact sets in IE ^d the corresponding optimal designs, i.e. probability measures with a certain extremal property, are concentrated ond(d+1)/2 points.     

环F2+uF2+vF2上自对偶码的两种构造方法

本文研究环R=F2+uF2+vF2上的自对偶码问题.利用Rn到F3n2的Gray映射及R上的自对偶码C的Gray像为F2上自对偶码,获得了R上任何偶长度的自对偶码存在性的结论.最后,给出了R上两种构造自对偶码的方法.     

On incidence energy of a graph

The Laplacian-energy like invariant LEL(G) and the incidence energy IE(G) of a graph are recently proposed quantities, equal, respectively, to the sum of the square roots of the Laplacian eigenvalues, and the sum of the singular values of the incidence matrix of the graph G. However, IE(G) is closely related with the eigenvalues of the Laplacian and signless Laplacian matrices of G. For bipartite graphs, IE=LEL. We now point out some further relations for IE and LEL: IE can be expressed in terms of eigenvalues of the line graph, whereas LEL in terms of singular values of the incidence matrix of a directed graph. Several lower and upper bounds for IE are obtained, including those that pertain to the line graph of G. In addition, Nordhaus-Gaddum-type results for IE are established.     

Semisymmetric graphs from polytopes

Every finite, self-dual, regular (or chiral) 4-polytope of type {3,q,3} has a trivalent 3-transitive (or 2-transitive) medial layer graph. Here, by dropping self-duality, we obtain a construction for semisymmetric trivalent graphs (which are edge- but not vertex-transitive). In particular, the Gray graph arises as the medial layer graph of a certain universal locally toroidal regular 4-polytope.     

The combinatorially regular polyhedra of index 2

Summary We investigate polyhedral realizations of regular maps with self-intersections in E³, whose symmetry group is a subgroup of index 2 in their automorphism group. We show that there are exactly 5 such polyhedra. The polyhedral sets have been more or less known for about 100 years; but the fact that they are realizations of regular maps is new in at least one case, a self-dual icosahedron of genus 11. Our polyhedra are closely related to the 5 regular compounds, which can be interpreted as discontinuous polyhedral realizations of regular maps.The author was born on March 5, 1937; so exactly half a century after Otto Haupt.Dedicated to Professor Otto Haupt with best wishes on his 100th birthday.     

Regular orbits of self-dual modules

We obtain a result regarding the existence of regular orbits for self-dual modules, thus generalizing a theorem of A. Espuelas for symplectic modules.

     

New Extremal Type I Codes of Lengths 40, 42, and 44

It is known that it is possible to construct a generator matrix for a self-dual code of length 2n+2 from a generator matrix of a self-dual code of length 2n. With the aid of a computer, we construct new extremal Type I codes of lengths 40, 42, and 44 from extremal self-dual codes of lengths 38, 40, and 42 respectively. Among them are seven extremal Type I codes of length 44 whose weight enumerator is 1+224y ⁸+872y ¹⁰+·. A Type I code of length 44 with this weight enumerator was not known to exist previously.     

Self-duality of rank 2 reflexive modules

Every finitely generated rank 2 third syzygy module over a regular local ring is known to be self-dual. We show more generally that any finitely generated rank 2 reflexive module is self-dual, and that the isomorphism is skew symmetric. We use this ismorphism to estimate how large k may be if the module is a kth syzygy, and how closely allied rank 3 modules are related to their duals.     

Cayley form,comass, and triality isomorphisms

Following an idea of Dadok, Harvey and Morgan, we apply the triality property of Spin(8) to calculate the comass of self-dual 4-forms on ℝ⁸. In particular, we prove that the Cayley form has comass 1 and that any self-dual 4-form realizing the maximal Wirtinger ratio (equation (1.5)) is SO(8)-conjugate to the Cayley form. We also use triality to prove that the stabilizer in SO(8) of the Cayley form is Spin(7). The results have applications in systolic geometry, calibrated geometry, and Spin(7) manifolds.     

A study on monotone self-dual Boolean functions

This paper shows that monotone self-dual Boolean functions in irredundant disjuntive normal form (IDNF) do not have more variables than disjuncts. Monotone self-dual Boolean functions in IDNF with the same number of variables and disjuncts are examined. An algorithm is proposed to test whether a monotone Boolean function in IDNF with n variables and n disjuncts is self-dual. The runtime of the algorithm is O(n³).     

Negacyclic self-dual codes over finite chain rings

In this article, we study negacyclic self-dual codes of length n over a finite chain ring R when the characteristic p of the residue field [`(R)]{\bar{R}} and the length n are relatively prime. We give necessary and sufficient conditions for the existence of (nontrivial) negacyclic self-dual codes over a finite chain ring. As an application, we construct negacyclic MDR self-dual codes over GR(p ^t , m) of length p ^m  + 1.     

Self-dual and quasi self-dual algebras

Self-dual algebras are ones with an A bimodule isomorphism A → A ^∨op, where A ^∨ = Hom _k (A, k) and A ^∨op is the same underlying k-module as A ^∨ but with left and right operations by A interchanged. These are in particular quasi self-dual algebras, i.e., ones with an isomorphism H*(A,A) ≌ H*(A,A ^{∨ op}). For all such algebras H*(A,A) is a contravariant functor of A. Finite-dimensional associative self-dual algebras over a field are identical with symmetric Frobenius algebras; an example of deformation of one is given. (The monoidal category of commutative Frobenius algebras is known to be equivalent to that of 1+1 dimensional topological quantum field theories.) All finite poset algebras are quasi self-dual.     

New constructions of optimal self-dual binary codes of length 54

The quaternary Hermitian self-dual [18,9,6]₄ codes are classified and used to construct new binary self-dual [54,27,10]₂ codes. All self-dual [54,27,10]₂ codes obtained have automorphisms of order 3, and six of their weight enumerators have not been previously encountered.      

For cones in Hilbert spaces, regularity is equivalent to self-duality

The theorem stating that a cone in a Hilbert space is regular if and only if it is self-dual is proved and applied to obtain new proofs of earlier results.Translated fromMatematicheskie Zametki, Vol. 64, No. 4, pp. 616–621, October, 1998.     

\mathbb{Z }_2\mathbb{Z }_4-Additive formally self-dual codes

We study odd and even \(\mathbb{Z }_2\mathbb{Z }_4\) formally self-dual codes. The images of these codes are binary codes whose weight enumerators are that of a formally self-dual code but may not be linear. Three constructions are given for formally self-dual codes and existence theorems are given for codes of each type defined in the paper.