A table of binary constant weight codes of length nl28 is presented. Explicit constructions are given for most of the 600 codes in the table; the majority of these codes are new. The known techniques for constructing constant weight codes are surveyed, and a table of (unrestricted) binary codes of length nl28 is given

/pdf/a-new-table-of-constant-weight-codes-7mqteex0ei.pdf

A new table of constant weight codes

This paper first considers several types of additive bases. A typical problem is to find $n_\gamma ( k )$, the largest n for which there exists a set $\{ 0 = a_1 < a_2 < \cdots < a_k \}$ of distinct integers modulo n such that each r in the range $0\leqq r \leqq n - 1$ can be written at least once as $r \equiv a_i + a_j $ (modulo n) with $i < j$. For example, $n_\gamma ( 8 ) = 24$, as illustrated by the set {0, 1, 2, 4, 8, 13, 18, 22}. The other problems arise if at least is changed to at most, or $i < j$ to $i\leqq j$, or if the words modulo n are omitted. Tables and bounds are given for each of these problems. Then a closely related graph labeling problem is studied. A connected graph with n edges is called harmonious if it is possible to label the vertices with distinct numbers (modulo n) in such a way that the edge sums are also distinct (modulo n). Some infinite families of graphs (odd cycles, ladders, wheels, $ \cdots $) are shown to be harmonious while others (even cycles, most complete or complete...

On Additive Bases and Harmonious Graphs

Let A(n,2\delta,w) denote the maximum number of codewords in any binary code of length n , constant weight w , and Hamming distance 2\delta Several lower bounds for A(n,2\delta,w) are given. For w and \delta fixed, A(n,2\delta,w) \geq n^{W-\delta+l}/w! and A(n,4,w)\sim n^{w-l}/w! as n \rightarrow \infty . In most cases these are better than the "Gilbert bound." Revised tables of A(n,2 \delta,w) are given in the range n \leq 24 and \delta \leq 5 .

Lower bounds for constant weight codes

We derive a new upper bound for the diameter of a k-regular graph G as a function of the eigenvalues of the adjacency matrix. Namely, suppose the adjacency matrix of G has eigenvalues AI, A2,)A,, with IA I I VI21 > *--> IA I where AI = k, A = IA21 . Then the diameter D(G) must satisfy D (G) We will consider families of graphs whose eigenvalues can be explicitly de- termined. These graphs are determined by sums or differences of vertex labels. Namely, the pair {i, j} being an edge depends only on the value i + j (or i j for directed graphs). We will show that these graphs are expander graphs with small diameters by using an inequality on character sums, which was recently proved by N. M. Katz. BELL COMMUNICATIONs RESEARCH, 445 SOUTH STREET, MORRISTOWN, NEW JERSEY 07960 This content downloaded from 207.46.13.111 on Sat, 21 May 2016 07:00:04 UTC All use subject to http://about.jstor.org/terms

/pdf/diameters-and-eigenvalues-pd23xa7xzl.pdf

Diameters and eigenvalues

(1.1) a1x1 + . . .+ akxk = b with x1, . . . , xk in a prescribed set of integers. We saw that the vanishing of the constant term b and the sum of coefficients s = a1 + . . . + ak had a strong effect on the behaviour of equation (1.1). The condition b = 0 is equivalent to homogeneity or multiplication invariance (if x1, . . . , xk is a solution, so is tx1, . . . , txk), while s = 0 means translation invariance (if x1, . . . , xk is a solution, so is x1 + t, . . . , xk+ t). We called equations with b = s = 0 invariant , and those with b 6= 0 or s 6= 0 noninvariant . In Part I of the paper we studied invariant equations; now we treat noninvariant ones. We recall the principal notations.

/pdf/solving-a-linear-equation-in-a-set-of-integers-i-1uzu3gd6o3.pdf

Solving a linear equation in a set of integers I

This paper extends some earlier results on difference sets andB2 sequences bySinger, Bose, Erdos andTuran, andChowla.

/pdf/theorems-in-the-additive-theory-of-numbers-3h06o7mtnz.pdf

Theorems in the additive theory of numbers

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Theorems in the additive theory of numbers