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

利用Jacobi和和相关的数论结果，证明了由AG（2n,3）中的2-平面构成的2-设计可以分解为比以前已知的分解更多的子设计。同时，通过检查环切分集中点的分布，也证明了对结果子设计数量的精确评估。本课题最初的目的是寻找可以应用于最优光学正交码的Steiner四重系统结构，但结果最终被证明适用于秘密共享方案和量子跃迁码。对于避免冲突的码，长度为n≡4（模数为8）、权值为3的最优码的直接构造是通过引入一个称为扩展奇序列的新概念来提供的。因此，根据先前已知的结果，长度和权值为3的最优避免冲突代码的大小范围已经完全确定。

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.