Related papers: There is no [21, 5, 14] code over F5

We consider extended $1$-perfect codes in Hamming graphs $H(n,q)$. Such nontrivial codes are known only when $n=2^k$, $k\geq 1$, $q=2$, or $n=q+2$, $q=2^m$, $m\geq 1$. Recently, Bespalov proved nonexistence of extended $1$-perfect codes for…

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.

The vertex Folkman number $F_v(s,t;k)$ is the smallest $n$ for which there exists a $K_k$-free graph on $n$ vertices whose vertices cannot be $2$-colored without producing a monochromatic copy of $K_s$ or $K_t$. We show $F_v(3,3;5)=8$. The…

Covering systems of the integers were introduced by Erd\H{o}s in 1950. Since then, many beautiful questions and conjectures about these objects have been posed. Most famously, Erd\H{o}s asked whether the minimum modulus of a covering system…

This paper has been withdrawn due to an error in the proof of the termination of the algorithm. Moreover, some ideas of this paper are a joint work with John Perry and it is inappropriate to mention only me as an author. John Perry was in…

In this paper we obtain the necessary condition for the existence of perfect $k$-colorings (equitable $k$-partitions) in Hamming graphs $H(n,q)$, where $q=2,3,4$ and Doob graphs $D(m,n)$. As an application, we prove the non-existence of…

We present a direct proof of the fact that the weak-$L^1$ and weak-$\ell^1$ spaces do not have type $1$.

We prove that a distance-regular graph with intersection array {56,36,9;1,3,48} does not exist. This intersection array is from the table of feasible parameters for distance-regular graphs in "Distance-regular graphs"\ by A.E. Brouwer, A.M.…

In this paper, we construct an infinite family of five-weight codes from trace codes over the ring $R=\mathbb{F}_2+u\mathbb{F}_2$, where $u^2=0.$ The trace codes have the algebraic structure of abelian codes. Their Lee weight is computed by…

Recently, linear codes with few weights have been widely studied, since they have applications in data storage systems, communication systems and consumer electronics. In this paper, we present a class of three-weight and five-weight linear…

In this paper we complete B\"{u}chi's proof that there is no decision algorithm for the solubility in integers of arbitrary systems of diagonal quadratic form equations, by proving the assertion that whenever $x_1^2, \cdots, x_5^2$ are five…

We prove that there is at least one irrationnal among the nine numbers zeta(5), zeta(7),..., zeta(21).

In this paper we show that there does not exist a strongly regular graph with parameters $(1911,270,105,27)$.

For a graph $G$, let $\chi(G)$ and $\omega(G)$ respectively denote the chromatic number and clique number of $G$. We give an explicit structural description of ($P_5$,gem)-free graphs, and show that every such graph $G$ satisfies…

In this note, we prove the irrationality of $\zeta(5)$ and generalize the method to prove the irrationality of all higher odd zeta values. Our proof relies on the method of contradiction, existence of solution of a system of Linear…

Flash memory is a non-volatile computer memory comprising blocks of cells, wherein each cell can take on q different values or levels. While increasing the cell level is easy, reducing the level of a cell can be accomplished only by erasing…

We show that no torus knot of type $(2,n)$, $n>3$ odd, can be obtained from a polynomial embedding $t \mapsto (f(t), g(t), h(t))$ where $(\deg(f),\deg(g))\leq (3,n+1) $. Eventually, we give explicit examples with minimal lexicographic…

In this paper, we consider the recently introduced concept of \emph{function-correcting codes (FCCs) with data protection}, which provide a certain level of error protection for the data and a higher level of protection for a desired…

Let $\mathbb{F}_{p^m}$ be a finite field of cardinality $p^m$ and $R=\mathbb{F}_{p^m}[u]/\langle u^2\rangle=\mathbb{F}_{p^m}+u\mathbb{F}_{p^m}$ $(u^2=0)$, where $p$ is an odd prime and $m$ is a positive integer. For any $\alpha,\beta\in…

In this note, we prove that every non-complete $(k+1)$-critical graph contains cycles of all lengths modulo $k$, where $k=4,5$.