Mathematical proof note · Version 1.0 · October 8, 2026
Exact Large-Alphabet Capacity of Aperiodic Frequency-Hopping Sequence Sets
National Mobile Communications Research Laboratory and the Frontiers Science Center for Mobile Information Communication and Security, Southeast University, Nanjing 211189, China
Abstract
We determine the maximum cardinality of aperiodic frequency-hopping sequence sets over sufficiently large alphabets. For each fixed length L, one threshold Q_0(L) works for every integer alphabet size q ≥ Q_0(L) and every correlation level 0 ≤ h<L: the capacity is ⌊ q^(h+1)/(L-h)⌋. The proof records every normalized set of matching positions, including nonconsecutive windows. A uniformly invertible window Gram operator permits positive degree regularization. A circular-word switch and a bounded-degree template give one sparse absorber for every divisible subset of reserved windows. A prescribed-vertex matching theorem then covers the bulk, and switching absorbs all but the exact integer remainder. The construction uses only alphabet symbols and therefore applies to integer alphabet sizes, including sizes that are not prime powers.
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Keywords
aperiodic frequency-hopping sequences; Hamming correlation; sequence-set capacity; large alphabets; hypergraph matching.
中文关键词：非周期跳频序列集；汉明相关；序列集容量；大字母表；超图匹配。
Suggested citation
S. Yang, "Exact Large-Alphabet Capacity of Aperiodic Frequency-Hopping Sequence Sets," mathematical proof note, version 1.0, Oct. 8, 2026. [Online]. Available: https://github.com/stalin1943/Coding-Theory-Proof-Notes/blob/main/aperiodic-frequency-hopping-capacity/proof.pdf