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The $q$-analogs of covering designs, Steiner systems, and Turán designs are studied. It is shown that $q$-covering designs and $q$-Turán designs are dual notions. A strong necessary condition for the existence of Steiner structures (the $q$-analogs of Steiner systems) over $\mathbb F$2 is given. No Steiner structures of strength $2$ or more are currently known, and our condition shows that their existence would imply the existence of new Steiner systems of strength $3$. The exact values of the $q$-covering numbers $\mathcal C$q$(n,k,1)$ and $\mathcal C$q$(n,n-1,r)$ are determined for all $q,n,k,r$. Furthermore, recursive upper and lower bounds on the size of general $q$-covering designs and $q$-Turán designs are presented. Finally, it is proved that $\mathcal C$2$(5,3,2) = 27$ and $\mathcal C$2$(7,3,2) \leq 399$. Tables of upper and lower bounds on $\mathcal C$2$(n,k,r)$ are given for all $n \leq 8$.
Improved binary constant weight codes with minimum distance 4 are constructed. A table with bounds on the chromatic number of small Johnson graphs is given.
keywords: Binary constant weight code.
Constant dimension codes, with a prescribed minimum distance, have found recently an application in network coding. All the codewords in such a code are subspaces of $\mathbb F$qn with a given dimension. A computer search for large constant dimension codes is usually inefficient since the search space domain is extremely large. Even so, we found that some constant dimension lexicodes are larger than other known codes. We show how to make the computer search more efficient. In this context we present a formula for the computation of the distance between two subspaces, not necessarily of the same dimension.
Let $A(n, d)$ denote the maximum number of codewords in a binary code of length n and minimum Hamming distance $d$. Upper and lower bounds on $A(n, d)$ have been a subject for extensive research. In this paper we examine upper bounds on $A(n, d)$ as a special case of bounds on the size of subsets in metric association scheme. We will first obtain general bounds on the size of such subsets, apply these bounds to the binary Hamming scheme, and use linear programming to further improve the bounds. We show that the sphere packing bound and the Johnson bound as well as other bounds are special cases of one of the bounds obtained from association schemes. Specific bounds on $A(n, d)$ as well as on the sizes of constant weight codes are also discussed.
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