AI 中文总结
研究外部差集族与代数操作检测码的关系，通过分圆构造得出有限域上外部部分差集族为EDFs的条件及偶数块大小的明确标准，还用多种方法构造有界广义强外部差集族，得到系统G - 最优强AMD码的无限族。
AI 中文摘要
外部差集族（EDFs）与代数操作检测（AMD）码密切相关，AMD码是一种密码原语，可保护消息免受无法观察传输码字的对手的加法篡改。我们首先研究有限域上外部部分差集族的分圆构造，并得出所得族为EDFs的充要条件。对于不超过14的偶数块大小，我们根据二次和双二次剩余得到明确标准，产生新的R - 最优弱AMD码。然后，我们使用有限域上的分圆类、有限域的直积和整数剩余环上的广义分圆类构造有界广义强外部差集族。这些构造给出了具有灵活参数的无限族系统G - 最优强AMD码。
英文摘要
External difference families (EDFs) are closely related to algebraic manipulation detection (AMD) codes, a cryptographic primitive that protects messages against additive tampering by an adversary who cannot observe the transmitted codeword. We first study a cyclotomic construction of external partial difference families over finite fields and derive a necessary and sufficient condition under which the resulting families are EDFs. For even block sizes up to $14$, we obtain explicit criteria in terms of quadratic and biquadratic residues, yielding new $R$-optimal weak AMD codes. We then construct bounded generalized strong external difference families using cyclotomic classes over finite fields, direct products of finite fields, and generalized cyclotomic classes over integer residue rings. These constructions give infinite families of systematic $G$-optimal strong AMD codes with flexible parameters.
