Related papers: Properties of perfect transitive binary codes of l…
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…
A complete classification of the perfect binary one-error-correcting codes of length 15 as well as their extensions of length 16 is presented. There are 5983 such inequivalent perfect codes and 2165 extended perfect codes. Efficient…
Studying binary perfect codes we show the existence of homogeneous nontransitive codes. Thus, as far as perfect codes are concerned, the propelinear codes are strictly contained in transitive codes, wheresas homogeneous codes form a strict…
The paper proves that there exist an exponential number of nonequivalent propelinear extended perfect binary codes of length growing to infinity. Specifically, it is proved that all transitive extended perfect binary codes found by Potapov…
A code is called transitive if its automorphism group (the isometry group) of the code acts transitively on its codewords. If there is a subgroup of the automorphism group acting regularly on the code, the code is called propelinear. Using…
It is shown that there does not exist a binary projective triply-even code of length $59$. This settles the last open length for projective triply-even binary codes. Therefore, projective triply-even binary codes exist precisely for lengths…
The present paper is devoted to study some completeness properties of transitive binary relational set, i.e., a set together with a transitive binary relation (so called t-set).
The aim of this work is a systematic investigation of the possible parameters of quasi-perfect (QP) binary and ternary linear codes of small dimensions and preparing a complete classification of all such codes. First we give a list of…
A complete classification of binary doubly even self-dual codes of length 40 is given. As a consequence, a classification of binary extremal self-dual codes of length 38 is also given.
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,…
We give a complete classification of binary linear complementary dual codes of lengths up to $13$ and ternary linear complementary dual codes of lengths up to $10$.
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…
Let $A_2(n,d)$ be the maximum size of a binary code of length $n$ and minimum distance $d$. In this paper we present the following new lower bounds: $A_2(18,4) \ge 5632$, $A_2(21,4) \ge 40960$, $A_2(22,4) \ge 81920$, $A_2(23,4) \ge 163840$,…
We show that there are good long binary generalized quasi-cyclic self-dual (either Type I or Type II) codes.
In this paper new infinite families of linear binary completely transitive codes are presented. They have covering radius $\rho = 3$ and 4, and are a half part of the binary Hamming and the binary extended Hamming code of length $n=2^m-1$…
From cosets of binary Hamming codes we construct diameter perfect constant-weight ternary codes with weight $n-1$ (where $n$ is the code length) and distances 3 and 5. The class of distance 5 codes has parameters unknown before. Keywords:…
We present a construction of 1-perfect binary codes, which gives a new lower bound on the number of such codes. We conjecture that this lower bound is asymptotically tight.
Given a parity-check matrix $H_m$ of a $q$-ary Hamming code, we consider a partition of the columns into two subsets. Then, we consider the two codes that have these submatrices as parity-check matrices. We say that anyone of these two…
We show that no projective 16-divisible binary linear code of length 131 exists. This implies several improved upper bounds for constant-dimension codes, used in random linear network coding, and partial spreads.
The binary Hamming codes with parameters $[2^m-1, 2^m-1-m, 3]$ are perfect. Their extended codes have parameters $[2^m, 2^m-1-m, 4]$ and are distance-optimal. The first objective of this paper is to construct a class of binary linear codes…