We consider the problem of computing the minimum length of functional batch and PIR codes of fixed dimension and for a fixed list size, over an arbitrary finite field. We recover, generalize, and refine several results that were previously obtained for binary codes. We present new upper and lower bounds for the minimum length, and discuss the asymptotic behaviour of this parameter. We also compute its value for several parameter sets. The paper also offers insights into the "correct" list size to consider for the Functional Batch Conjecture over non-binary finite fields, and establishes various supporting results.

翻译：本文研究了在任意有限域上，针对固定维度和固定列表大小，计算函数批处理码与PIR码最小长度的问题。我们恢复、推广并改进了先前在二进制码研究中获得的若干结果。针对最小长度，我们提出了新的上界与下界，并讨论了该参数的渐近行为。同时，我们对多组参数集计算了该参数的具体数值。本文还探讨了在非二进制有限域上，函数批处理猜想应考虑的“恰当”列表大小，并建立了若干支撑性结论。