Constructions of optimal optical orthogonal codes and conflict-avoiding codes derived from combinatorial designs

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By applying Jacobi sums and some related number theoretic results, it is shown that the 2-design formed by the 2-flats in AG(2n,3) can be decomposed into more subdesigns than a previously known decomposition. At the same time, exact evaluation of the number of the resulting subdesigns is also demonstrated by examining the distribution of points in cyclotomic cosets. The original purpose of this theme was to find constructions of Steiner quadruple systems which can be applied to optimal optical orthogonal codes, but the result eventually turned out to be applicable to secret sharing scheme and quantum jump codes. As for conflict-avoiding codes, direct constructions for optimal codes of length n≡4 (mod 8) and weight 3 are provided by bringing in a new concept called an extended odd sequence. As a consequence, with previously known results, the spectrum of the size of optimal conflict-avoiding codes of even length and weight 3 is completely settled.