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The affine linear group of degree one, AGL (1 , F q) , over the finite field F q , acts sharply two-transitively on F q . Given S < AGL (1 , F q) and an integer k, 1 ≤ k ≤ q , does there exist a k-element subset B ⊂ F q whose set-wise stabilizer is S? Our main result is the derivation of two formulas which provide an answer to this question. This result allows us to determine all possible parameters of binary constant weight codes that are constructed from the action of AGL (1 , F q) on F q to meet the Johnson bound. Consequently, for many parameters, we are able to determine the values of the function A 2 (n , d , w) , which is the maximum number of codewords in a binary constant weight code of length n, weight w and minimum distance ≥ d . [ABSTRACT FROM AUTHOR]

Wang and Yin have established the existence of a directed BIBD with block size 7 and index 1 (a DBIBD( v, 7, 1)) for all v ≡ 1, 7 (mod 21) except for v = 22 and possibly for 68 other cases. In this paper, we reduce the number of possible exceptions to 4, namely v = 274, 358, 400, 526. Correspondingly, for all such v, our results establish the existence of a T(2, 7, v)-code or equivalently a perfect 5-deletion-correcting code with words of length 7 over an alphabet of size v, where all the coordinates must be different. In the process, we also reduce the possible exceptions for ( v, 7, 2)-BIBDs to 2 cases, v = 274 and 358 (in addition to the non-existent (22, 7, 2)-BIBD). [ABSTRACT FROM AUTHOR]

The smallest BIBD, as for the number of points and blocks, whose existence is still undecided is 2-(22, 8, 4). Possible subconfigurations of such a design, namely 2-(10, 4, 4) designs, are here ruled out. The result is obtained by classifying all 2-(10, 4, 4) designs and trying to find 2-(22, 8, 4) designs by solving instances of the maximum clique problem. [ABSTRACT FROM AUTHOR]

We consider 2-(9, 3, λ) designs, which are known to exist for all λ ≥ 1, andenumerate such designs for λ = 5 and their resolutions for 3 ≤ λ ≤ 5, the smallestopen cases. The number of nonisomorphic such structures obtained is 5 862 121 434, 426, 149 041, and 203 047732, respectively. The designs are obtained by an orderly algorithm, and the resolutions by two approaches:either by starting from the enumerated designs and applying a clique-finding algorithm on two levels or by anorderly algorithm. [ABSTRACT FROM AUTHOR]

The necessary conditions for the existence of a balanced incomplete block design on v points, with index λ and block size k, are that: In this paper we study k=9 with λ=2,4 or 8. For λ=8, we show these conditions on v are sufficient, and for λ=2, 4 respectively there are 8 and 3 possible exceptions the largest of which are v=1845 and 783. We also give some examples of group divisible designs derived from balanced ternary designs. [ABSTRACT FROM AUTHOR]

The object of this paper is the construction of balanced incomplete block designs with k=7. This paper continues the work begun by Hanani, who solved the construction problem for designs with a block size of 7, and with λ=6, 7, 21 and 42. The construction problem is solved here for designs with λ > 2 except for v=253, λ= 4,5 ; also for λ= 2, the number of unconstructed designs is reduced to 9 (1 nonexistent, 8 unknown). [ABSTRACT FROM AUTHOR]

The necessary condition for the existence of a resolvable balanced incomplete block design on v points, with λ = 1 and k = 8, is that v ≡ 8 mod 56. With the exception of 66 values of v, this condition is shown to be sufficient. The largest exceptional value of v is 24480. [ABSTRACT FROM AUTHOR]