For $$n,d,w \in \mathbb {N}$$ n , d , w ∈ N , let A(n, d, w) denote the maximum size of a binary code of word length n, minimum distance d and constant weight w. Schrijver recently showed using semidefinite programming that $$A(23,8,11)=1288$$ A ( 23 , 8 , 11 ) = 1288 , and the second author that $$A(22,8,11)=672$$ A ( 22 , 8 , 11 ) = 672 and $$A(22,8,10)=616$$ A ( 22 , 8 , 10 ) = 616 . Here we show uniqueness of the codes achieving these bounds. Let A(n, d) denote the maximum size of a binary code of word length n and minimum distance d. Gijswijt et al. showed that $$A(20,8)=256$$ A ( 20 , 8 ) = 256 . We show that there are several nonisomorphic codes achieving this bound, and classify all such codes with all distances divisible by 4.

We study the monoid generated by $$n \times n$$ distance matrices under tropical (or min-plus) multiplication. Using the tropical geometry of the orthogonal group, we prove that this monoid is a finite polyhedral fan of dimension $$\left( {\begin{array}{c}n\\ 2\end{array}}\right) $$ , and we compute the structure of this fan for $$n$$ up to $$5$$ . The monoid captures gossip among $$n$$ gossipers over lossy phone lines, and contains the gossip monoid over ordinary phone lines as a submonoid. We prove several new results about this submonoid as well. In particular, we establish a sharp bound on chains of calls in each of which someone learns something new.

We characterize a 55-point graph by its spectrum $${4^1, (-2)^{10}, (-1 \pm \sqrt{3})^{10}, ((3 \pm \sqrt{5})/2)^{12}}$$ . No interlacing is used: examination of tr A m for m ≤ 7 together with study of the representation in the eigenspace for the eigenvalue −2 suffices.

We show that the q-Kneser graph qK 2k:k (the graph on the k-subspaces of a 2k-space over GF(q), where two k-spaces are adjacent when they intersect trivially), has chromatic number q k + q k−1 for k = 3 and for k < q log q − q. We obtain detailed results on maximal cocliques for k = 3.

Consider the partial linear space on the images in Λ /2 Λ of the shortest nonzero vectors in the Leech lattice Λ , where the lines are the triples of vectors adding up to zero. We determine the universal embedding dimension and the generating rank of this space (both are 24) and classify its hyperplanes.

We give some necessary conditions for a graph to be 3-chromatic in terms of the spectrum of the adjacency matrix. For all known distance-regular graphs it is determined whether they are 3-chromatic. A start is made with the classification of 3-chromatic distance-regular graphs, and it is shown that such graphs, if not complete 3-partite, must have λ ≤ 1.

An associative block design is a certain balanced partition of a hypercube into smaller hypercubes. We construct such a design, thus settling the smallest open case.

Upper and lower bounds are presented for the maximal possible size of mixed binary/ternary error-correcting codes. A table up to length 13 is included. The upper bounds are obtained by applying the linear programming bound to the product of two association schemes. The lower bounds arise from a number of different constructions.

Combining Delsarte's (1973) linear programming bound with the information that certain weights cannot occur, new upper bounds for d/sub min/ (n,k), the maximum possible minimum distance of a binary linear code with given word length n and dimension k, are derived.>

Tables with lower and upper bounds for d/sub max/ (n,k), the maximum possible minimum distance of a binary linear code with word length n and dimension k, are shown.>

A table of binary constant weight codes of length n>

The question of the uniqueness of a certain combinatorial structure has arisen in three contexts: a) is the regular near octagon with parameters(s,t_{2},t_{3},t)=(1, 1,2,23)unique [5]? b) is the partial2-geometry with nexus three and blocksize24unique [2]? c) is there a unique graph such that it is the graph of a parallelism of\left(^{24}_{4}\right)with respect to any [1]? We observe that these questions are equivalent and give an affirmative answer. In fact, we prove a more general theorem, showing the truth of a conjecture by Cameron.

A method for searching for constant weight codes is described and used to construct 14 codes which are better than the known codes with the same parameters.

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Improved bounds forA(n,d), the maximum number of codewords in a (linear or nonlinear) binary code of word lengthnand minimum distanced, and forA(n,d,w), the maximum number of binary vectors of lengthn, distanced, and constant weightwin the rangen \leq 24andd \leq 10are presented. Some of the new values areA