Related papers: On perfect codes that do not contain Preparata-lik…
The Fibonacci cube of dimension n, denoted as $\Gamma$ n , is the subgraph of the n-cube 5 Q n induced by vertices with no consecutive 1's. Ashrafi and his co-authors proved the non-existence of perfect codes in $\Gamma$ n for n $\ge$ 4. As…
We study the problem of existence of (nontrivial) perfect codes in the discrete $ n $-simplex $ \Delta_{\ell}^n := \left\{ \begin{pmatrix} x_0, \ldots, x_n \end{pmatrix} : x_i \in \mathbb{Z}_{+}, \sum_i x_i = \ell \right\} $ under $ \ell_1…
Lee codes have been intensively studied for more than 40 years. Interest in these codes has been triggered by the Golomb-Welch conjecture on the existence of the perfect error-correcting Lee codes. In this paper we deal with the existence…
A cross-bifix-free code is a set of words in which no prefix of any length of any word is the suffix of any word in the set. Cross-bifix-free codes arise in the study of distributed sequences for frame synchronization. We provide a new…
Ternary constant weight codes of length $n=2^m$, weight $n-1$, cardinality $2^n$ and distance $5$ are known to exist for every $m$ for which there exists an APN permutation of order $2^m$, that is, at least for all odd $m \geq 3$ and for…
The {\em Fibonacci cube} of dimension $n$, denoted as $\Gamma\_n$, is the subgraph of the $n$-cube $Q\_n$ induced by vertices with no consecutive 1's. In an article of 2016 Ashrafi and his co-authors proved the non-existence of perfect…
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 define a variable-length code having the property that no (non-empty) prefix of each its codeword is a suffix of any other one, and vice versa. This kind of code can be seen as an extension of two well-known codes in literature, called…
A variable-length code is a fix-free code if no codeword is a prefix or a suffix of any other codeword. In a fix-free code any finite sequence of codewords can be decoded in both directions, which can improve the robustness to channel noise…
We classify all binary error correcting completely regular codes of length $n$ with minimum distance $\delta>n/2$.
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…
Perfect codes obtained by the Vasil'ev--Sch\"onheim construction from a linear base code and quadratic switching functions are transitive and, moreover, propelinear. This gives at least $\exp(cN^2)$ propelinear $1$-perfect codes of length…
We study $1$-perfect codes in Doob graphs $D(m,n)$. We show that such codes that are linear over $GR(4^2)$ exist if and only if $n=(4^{g+d}-1)/3$ and $m=(4^{g+2d}-4^{g+d})/6$ for some integers $g \ge 0$ and $d>0$. We also prove necessary…
We study codes with parameters of the ternary Hamming $(n=(3^m-1)/2,3^{n-m},3)$ code, i.e., ternary $1$-perfect codes. The rank of the code is defined to be the dimension of its affine span. We characterize ternary $1$-perfect codes of rank…
A complete classification of the perfect binary one-error-correcting codes of length 15 as well as their extensions of length 16 was recently carried out in [P. R. J. \"Osterg{\aa}rd and O. Pottonen, "The perfect binary one-error-correcting…
The doubly shortened perfect codes of length 13 are classified utilizing the classification of perfect codes in [P.R.J. \"Osterg{\aa}rd and O. Pottonen, The perfect binary one-error-correcting codes of length 15: Part I - Classification,…
In this survey we concern ourself with the question, wether there exists a fix-free code for a given sequence of codeword lengths. For a given alphabet, we obtain the {\em Kraftsum} of a code, if we divide for every length the number of…
This paper provides a new instance of quantum deletion error-correcting codes. This code can correct any single quantum deletion error, while our code is only of length 4. This paper also provides an example of an encoding quantum circuit…
In this note, we study the classification of $\mathbb{Z}_4$-codes. For some special cases $(k_1,k_2)$, by hand, we give a classification of $\mathbb{Z}_4$-codes of length $n$ and type $4^{k_1}2^{k_2}$ satisfying a certain condition. Our…
Delsarte conjectured in 1973 that there are no nontrivial pefect codes in the Johnson scheme. Etzion and Schwartz recently showed that perfect codes must be k-regular for large k, and used this to show that there are no perfect codes…