相关论文: Non-existence of a ternary constant weight $(16, 5…
All codes with minimum distance 8 and codimension up to 14 and all codes with minimum distance 10 and codimension up to 18 are classified. Nonexistence of codes with parameters [33,18,8] and [33,14,10] is proved. This leads to 8 new exact…
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…
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…
A construction of 2-quasi-perfect Lee codes is given over the space $\mathbb Z_p^n$ for $p$ prime, $p\equiv \pm 5\pmod{12}$ and $n=2[\frac{p}{4}]$. It is known that there are infinitely many such primes. Golomb and Welch conjectured that…
In this paper, for an even integer $n\geq 4$ and any positive integer $k$ with ${\rm gcd}(n/2,k)={\rm gcd}(n/2-k,2k)=d$ being odd, a class of $p$-ary codes $\mathcal{C}^k$ is defined and their weight distribution is completely determined,…
In an $[n,k,d]$ linear code, a code symbol is said to have locality $r$ if it can be repaired by accessing at most $r$ other code symbols. For an $(n,k,r)$ \emph{locally repairable code} (LRC), the minimum distance satisfies the well-known…
We show an interesting PBD-closure result for the set of lengths of constant-composition codes whose distance and size meet certain conditions. A consequence of this PBD-closure result is that the size of optimal constant-composition codes…
The Johnson-type upper bound on the maximum size of a code of length $n$, distance $d=2w-1$ and constant composition ${\overline{w}}$ is $\lfloor\dfrac{n}{w_1}\rfloor$, where $w$ is the total weight and $w_1$ is the largest component of…
We develop and apply combinatorial algorithms for investigation of the feasible distance distributions of binary orthogonal arrays with respect to a point of the ambient binary Hamming space utilizing constraints imposed from the relations…
For nonnegative integers $n,d,w$, let $A(n,d,w)$ be the maximum size of a code $C \subseteq \mathbb{F}_2^n$ with constant weight $w$ and minimum distance at least $d$. We consider two semidefinite programs based on quadruples of code words…
A well known conjecture of Golomb and Welch is that the only nontrivial perfect codes in the Lee and Manhattan metrics have length two or minimum distance three. This problem and related topics were subject for extensive research in the…
We study trace codes with defining set $L,$ a subgroup of the multiplicative group of an extension of degree $m$ of the alphabet ring $\mathbb{F}_3+u\mathbb{F}_3+u^{2}\mathbb{F}_{3},$ with $u^{3}=1.$ These codes are abelian, and their…
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…
Best and Brouwer [Discrete Math. 17 (1977), 235-245] proved that triply-shortened and doubly-shortened binary Hamming codes (which have length $2^m-4$ and $2^m-3$, respectively) are optimal. Properties of such codes are here studied,…
The concept of group divisible codes, a generalization of group divisible designs with constant block size, is introduced in this paper. This new class of codes is shown to be useful in recursive constructions for constant-weight and…
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…
We classify binary completely regular codes of length $m$ with minimum distance $\delta$ for $(m,\delta)=(12,6)$ and $(11,5)$. We prove that such codes are unique up to equivalence, and in particular, are equivalent to certain Hadamard…
A new construction for constant weight codes is presented. The codes are constructed from $k$-dimensional subspaces of the vector space $\F_q^n$. These subspaces form a constant dimension code in the Grassmannian space $\cG_q(n,k)$. Some of…
We consider the problem of finding $A_2(n,\{d_1,d_2\})$ defined as the maximal size of a binary (non-linear) code of length $n$ with two distances $d_1$ and $d_2$. Binary codes with distances $d$ and $d+2$ of size…
A perfect number is a positive integer $N$ such that the sum of all the positive divisors of $N$ equals $2N$, denoted by $\sigma(N) = 2N$. The question of the existence of odd perfect numbers (OPNs) is one of the longest unsolved problems…