Related papers: A New Lower Bound for A(17,6,6)
Let $C$ be a binary code of length $n$ with distances $0<d_1<\cdots<d_s\le n$. In this note we prove a general upper bound on the size of $C$ without any restriction on the distances $d_i$. The bound is asymptotically optimal.
Upper and lower bounds are derived for the quantity in the title, which is tabulated for modest values of $n$ and $k.$ An application to graphs with many cycles is given.
For $(n,d)= (66,17),(78,19)$ and $(94,21)$, we construct quantum $[[n,0,d]]$ codes which improve the previously known lower bounds on the largest minimum weights among quantum codes with these parameters. These codes are constructed from…
In this paper, we give a new upper bound on the minimum Euclidean weight of Type II $\ZZ_{2k}$-codes and the concept of extremality for the Euclidean weights when $k=3,4,5,6$. Together with the known result, we demonstrate that there is an…
We show that $A_2(7,4) \leq 388$ and, more generally, $A_q(7,4) \leq (q^2-q+1)[7]_q + q^4 - 2q^3 + 3q^2 - 4q + 4$ by semidefinite programming for $q \leq 101$. Furthermore, we extend results by Bachoc et al. on SDP bounds for $A_2(n,d)$,…
We propose new results on low weight codewords of affine and projective generalized Reed-Muller codes. In the affine case we prove that if the size of the working finite field is large compared to the degree of the code, the low weight…
We derive a new estimate of the size of finite sets of points in metric spaces with few distances. The following applications are considered: (1) we improve the Ray-Chaudhuri--Wilson bound of the size of uniform intersecting families of…
If the list of binary numbers is read by upward-sloping diagonals, the resulting ``sloping binary numbers'' 0, 11, 110, 101, 100, 1111, 1010, ... (or 0, 3, 6, 5, 4, 15, 10, ...) have some surprising properties. We give formulae for the n-th…
In this work, we define a modification of a bordered construction for self-dual codes which utilises $\lambda$-circulant matrices. We provide the necessary conditions for the construction to produce self-dual codes over finite commutative…
Using the method for constructing binary self-dual codes with an automorphism of order square of a prime number we have classified all binary self-dual codes with length 76 having minimum distance $d=14$ and automorphism of order 9. Up to…
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…
Generalized bicycle (GB) codes is a class of quantum error-correcting codes constructed from a pair of binary circulant matrices. Unlike for other simple quantum code ans\"atze, unrestricted GB codes may have linear distance scaling. In…
In this paper, we determine the dimensions of the residue codes of extremal Type II Z4-codes for lengths 32 and 40. We demonstrate that every binary doubly even self-dual code of length 32 can be realized as the residue code of some…
The concepts of pseudocodeword and pseudoweight play a fundamental role in the finite-length analysis of LDPC codes. The pseudoredundancy of a binary linear code is defined as the minimum number of rows in a parity-check matrix such that…
Let $n (>3)$ be a prime number and $\Bbb F_{2^n}$ a finite field of $2^n$ elements. Let $L =\Bbb F_{2^n}\cup \{\infty\}$ be the support set and $g(x)$ an irreducible polynomial of degree $6$ over $\Bbb F_{2^n}$. In this paper, we obtain an…
This paper considers $n= 128$ dimensional construction A lattice design, using binary codes with known minimum Hamming distance and codeword multiplicity, the number of minimum weight codeword. A truncated theta series of the lattice is…
We give a classification of four-circulant singly even self-dual $[60,30,d]$ codes for $d=10$ and $12$. These codes are used to construct extremal singly even self-dual $[60,30,12]$ codes with weight enumerator for which no extremal singly…
The deletion distance between two binary words $u,v \in \{0,1\}^n$ is the smallest $k$ such that $u$ and $v$ share a common subsequence of length $n-k$. A set $C$ of binary words of length $n$ is called a $k$-deletion code if every pair of…
In this paper we present several classes of asymptotically good concatenated quantum codes and derive lower bounds on the minimum distance and rate of the codes. We compare these bounds with the best-known bound of…
We consider a new family of codes, termed asymmetric Lee distance codes, that arise in the design and implementation of DNA-based storage systems and systems with parallel string transmission protocols. The codewords are defined over a…