The build-up construction of quasi self-dual codes over a non-unital ring

There is a local ring [Formula: see text] of order [Formula: see text] without identity for the multiplication, defined by generators and relations as [Formula: see text] We study a recursive construction of self-orthogonal codes over [Formula: see text] We classify, up to permutation equivalence, self-orthogonal codes of length [Formula: see text] and size [Formula: see text] (called here quasi self-dual codes or QSD) up to the length [Formula: see text]. In particular, we classify Type IV codes (QSD codes with even weights) up to [Formula: see text].

On Graph-Orthogonal Arrays by Mutually Orthogonal Graph Squares

During the last two centuries, after the question asked by Euler concerning mutually orthogonal Latin squares (MOLS), essential advances have been made. MOLS are considered as a construction tool for orthogonal arrays. Although Latin squares have numerous helpful properties, for some factual applications these structures are excessively prohibitive. The more general concepts of graph squares and mutually orthogonal graph squares (MOGS) offer more flexibility. MOGS generalize MOLS in an interesting way. As such, the topic is attractive. Orthogonal arrays are essential in statistics and are related to finite fields, geometry, combinatorics and error-correcting codes. Furthermore, they are used in cryptography and computer science. In this paper, our current efforts have concentrated on the definition of the graph-orthogonal arrays and on proving that if there are k MOGS of order n, then there is a graph-orthogonal array, and we denote this array by G-OA(n2,k,n,2). In addition, several new results for the orthogonal arrays obtained from the MOGS are given. Furthermore, we introduce a recursive construction method for constructing the graph-orthogonal arrays.

A Recursive Construction for Skew Hadamard Difference Sets

A major conjecture on the existence of abelian skew Hadamard difference sets is: if an abelian group $G$ contains a skew Hadamard difference set, then $G$ must be elementary abelian. This conjecture remains open in general. In this paper, we give a recursive construction for skew Hadamard difference sets in abelian (not necessarily elementary abelian) groups. The new construction can be considered as a result on the aforementioned conjecture: if there exists a counterexample to the conjecture, then there exist infinitely many counterexamples to it.

Arrangements of Pseudocircles: Triangles and Drawings

A pseudocircle is a simple closed curve on the sphere or in the plane. The study of arrangements of pseudocircles was initiated by Grünbaum, who defined them as collections of simple closed curves that pairwise intersect in exactly two crossings. Grünbaum conjectured that the number of triangular cells $$p_3$$ p 3 in digon-free arrangements of n pairwise intersecting pseudocircles is at least $$2n-4$$ 2 n - 4 . We present examples to disprove this conjecture. With a recursive construction based on an example with 12 pseudocircles and 16 triangles we obtain a family of intersecting digon-free arrangements with $$p_3({\mathscr {A}})/n \rightarrow 16/11 = 1.\overline{45}$$ p 3 ( A ) / n → 16 / 11 = 1 . 45 ¯ . We expect that the lower bound $$p_3({\mathscr {A}}) \ge 4n/3$$ p 3 ( A ) ≥ 4 n / 3 is tight for infinitely many simple arrangements. It may however be true that all digon-free arrangements of n pairwise intersecting circles have at least $$2n-4$$ 2 n - 4 triangles. For pairwise intersecting arrangements with digons we have a lower bound of $$p_3 \ge 2n/3$$ p 3 ≥ 2 n / 3 , and conjecture that $$p_3 \ge n-1$$ p 3 ≥ n - 1 . Concerning the maximum number of triangles in pairwise intersecting arrangements of pseudocircles, we show that $$p_3 \le \frac{4}{3}\left( {\begin{array}{c}n\\ 2\end{array}}\right) +O(n)$$ p 3 ≤ 4 3 n 2 + O ( n ) . This is essentially best possible because there are families of pairwise intersecting arrangements of n pseudocircles with $$p_3 = \frac{4}{3}\left( {\begin{array}{c}n\\ 2\end{array}}\right) $$ p 3 = 4 3 n 2 . The paper contains many drawings of arrangements of pseudocircles and a good fraction of these drawings was produced automatically from the combinatorial data produced by our generation algorithm. In the final section we describe some aspects of the drawing algorithm.

Discrete Orthogonality of Bivariate Polynomials of A2, C2 and G2

We develop discrete orthogonality relations on the finite sets of the generalized Chebyshev nodes related to the root systems A 2 , C 2 and G 2 . The orthogonality relations are consequences of orthogonality of four types of Weyl orbit functions on the fragments of the dual weight lattices. A uniform recursive construction of the polynomials as well as explicit presentation of all data needed for the discrete orthogonality relations allow practical implementation of the related Fourier methods. The polynomial interpolation method is developed and exemplified.