engleski

Equations of tactical decomposition of designs: application to the problem of existence of 3- (16, 7, 5) design

We address tactical decomposition of t- (v, k, lambda_t) designs. Equations for coefficients of tactical decomposition matrices when t=2 are well-known. In this talk, we generalize these equations and propose an explicit equation system for coefficients of tactical decomposition matrices for t- (v, k, lambda_t) designs, for any integer value of t. This system of equations for coefficients of tactical decomposition matrices represents necessary conditions for the existence of t- designs with an assumed automorphism group. The problem of existence of a 3-(16, 7, 5) design is one of the open problems in design theory. These parameters describe one of the smallest 3- designs for which the question of existence is still unsolved. Previous results show that if a 3- (16, 7, 5) design admits an automorphism of a prime order $p$, then p = 2, 3. We use equations of tactical decomposition to eliminate the case p = 3.

combinatorial design ; automorphism group ; tactical decomposition

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