Related papers: Nonexistence for extremal Type II $\ZZ_{2k}$-Codes

We consider $q$-ary (linear and nonlinear) block codes with exactly two distances: $d$ and $d+\delta$. Several combinatorial constructions of optimal such codes are given. In the linear (but not necessary projective) case, we prove that…

Using integer linear programming and table-lookups we prove that there is no binary linear $[1988, 12, 992]$ code. As a by-product, the non-existence of binary linear codes with the parameters $[324, 10, 160]$, $[356, 10, 176]$,…

This research announcement describes in very rough terms methods and a computer language under development, which can be used to prove the nonexistence of binary linear codes. Over a hundred new results have been obtained by the author. For…

We prove that $K_{n}$ is $\mathbb{Z}_{2}^{2}$-cordial if and only if $1 \leq n \leq 3$ and that $K_{m,n}$ is $\mathbb{Z}_{2}^{2}$ if and only if it is false that $m=n=2$.

In this work, quadratic double and quadratic bordered double circulant constructions are applied to F_4 + uF_4 as well as F_4, as a result of which extremal binary self-dual codes of length 56 and 64 are obtained. The binary extension…

The Golomb-Welch conjecture (1968) states that there are no $e$-perfect Lee codes in $\mathbb{Z}^n$ for $n\geq 3$ and $e\geq 2$. This conjecture remains open even for linear codes. A recent result of Zhang and Ge establishes the…

Professor Cunsheng Ding gave cyclotomic constructions of cyclic codes with length being the product of two primes. In this paper, we study the cyclic codes of length $n=2^e$ and dimension $k=2^{e-1}$. Clearly, Ding's construction is not…

We prove that the only primes which may divide the order of the automorphism group of a putative binary self-dual doubly-even [120, 60, 24] code are 2, 3, 5, 7, 19, 23 and 29. Furthermore we prove that automorphisms of prime order $p \geq…

Let $P_{2,2}$ be the orientation of $C_4$ which consists of two 2-paths with the same initial and terminal vertices. In this paper, we determine the maximum size of $P_{2,2}$-free digraphs of order $n$ as well as the extremal digraphs…

In this paper we construct a cover {a_s(mod n_s)}_{s=1}^k of Z with odd moduli such that there are distinct primes p_1,...,p_k dividing 2^{n_1}-1,...,2^{n_k}-1 respectively. Using this cover we show that for any positive integer m divisible…

Currently, the existence of an extremal singly even self-dual code of length $24k+10$ is unknown for all nonnegative integers $k$. In this note, we study singly even self-dual $[24k+10,12k+5,4k+2]$ codes. We give some restrictions on the…

In this work, we introduce new construction methods for self-dual codes using a Baumert-Hall array. We apply the constructions over the alphabets F_2 and F_4 + uF_4 and combine them with extension theorems and neighboring constructions. As…

We show that for every length of form $4^k-1$, there exists a binary $1$-perfect code that does not include any Preparata-like code.

Supernovae have been confirmed to redshift z ~ 1.7 for type Ia (thermonuclear detonation of a white dwarf) and to z ~ 0.7 for type II (collapse of the core of the star). The subclass type IIn supernovae are luminous core-collapse explosions…

We give a new proof of the fact that Barker polynomials of even degree greater than 12, and hence Barker sequences of odd length greater than 13 do not exist. This is intimately tied to irreducibility questions and proved as a consequence…

Binary sequences with lower autocorrelation values have important applications in cryptography and communications. In this paper, we present all possible parameters for binary periodical sequences with a 2-level autocorrelation values. For…

We address the maximum size of binary codes and binary constant weight codes with few distances. Previous works established a number of bounds for these quantities as well as the exact values for a range of small code lengths. As our main…

We study the construction of quasi-cyclic self-dual codes, especially of binary cubic ones. We consider the binary quasi-cyclic codes of length 3\ell with the algebraic approach of [9]. In particular, we improve the previous results by…

We prove that a non-affine latin quandle (also known as left distributive quasigroup) of order $2^k$ exists if and only if $k = 6$ or $k \geq 8$. The construction is expressed in terms of central extensions of affine quandles.

From a given $[n, k]$ code $C$, we give a method for constructing many $[n, k]$ codes $C'$ such that the hull dimensions of $C$ and $C'$ are identical. This method can be applied to constructions of both self-dual codes and linear…