Design of Balanced Wide Gap No-Hit Zone Sequences with Optimal Auto-Correlation

Frequency-hopping multiple access is widely adopted to blunt narrow-band jamming and limit spectral disclosure in cyber–physical systems, yet its practical resilience depends on three sequence-level properties. First, balancedness guarantees that every carrier is occupied equally often, removing spectral peaks that a jammer or energy detector could exploit. Second, a wide gap between successive hops forces any interferer to re-tune after corrupting at most one symbol, thereby containing error bursts. Third, a no-hit zone (NHZ) window with a zero pairwise Hamming correlation eliminates user collisions and self-interference when chip-level timing offsets fall inside the window. This work introduces an algebraic construction that meets the full set of requirements in a single framework. By threading a permutation over an integer ring and partitioning the period into congruent sub-blocks tied to the desired NHZ width, we generate balanced wide gap no-hit zone frequency-hopping (WG-NHZ FH) sequence sets. Analytical proofs show that (i) each sequence achieves the Lempel–Greenberger bound for auto-correlation, (ii) the family and zone sizes satisfy the Ye–Fan bound with equality, (iii) the hop-to-hop distance satisfies a provable WG condition, and (iv) balancedness holds exactly for every carrier frequency.