New Families of Optimal Frequency-Hopping Sequences of Composite Lengths

Abstract

Frequency-hopping sequences (FHSs) are employed to mitigate the interferences caused by the hits of frequencies in frequency-hopping spread spectrum systems. In this paper, we present two new constructions for FHS sets. We first give a new construction for FHS sets of length nN for two positive integers n and N with gcd(n, N ) = 1. We then present another construction for FHS sets of length (q − 1)N , where q is a prime power satisfying gcd(q − 1, N ) = 1. By these two constructions, we obtain infinitely many new optimal FHS sets with respect to the Peng-Fan bound as well as new optimal FHSs with respect to the Lempel-Greenberger bound, which have length nN or n(q − 1)N . As a result, a great deal of flexibility may be provided in the choice of FHS sets for a given frequency-hopping spread spectrum system.

New Families of Optimal Frequency Hopping Sequence Sets

“…From these constructions, we obtained many infinitely families of new optimal FHS sets with respect to the Peng-Fan bound (4). Our combinatorial constructions generalized the previous methods, the recursive construction for BNCDPs in [16] became a special case of Theorem 4.7, the constraint gcd(w, n) = 1 of Construction A in [4] was removed, the existence proof of a (v, f, e; v−1 e + 1)-FHS set in [23] was simplified by using cyclotomic cosets. Compared with the previous extension methods in [4], [5], [16] and [22], our recursive constructions gave new optimal FHS sets for much more general cases.…”

“…, applying Theorem 4.7 yields a (3p 1 · · · p r+1 , 3p 2 · · · p r+1 , {{4}, {4}}, 4)-BNCRDP of size 3p1···pr+1−3p2···pr+1 4 over Z 3p1···pr+1 such that each CRDP is a partition of Z 3p1···pr+1 \ p 1 Z 3p1···pr+1 . By induction hypothesis there exits a partition-type (3p 2 · · · p r+1 , {{3, 4}, {3, 4}}, 4)-BNCDP over Z 3p2···pr+1 of size 3p2···pr+1 4 , applying Lemma 4.6 then yields a partition-type (3p 1 · · · p r+1 , {{3, 4}, {3, 4}}, 4)-BNCDP of size 3p1···pr+1+1 4 over Z 3p1···pr+1 .…”

New Extension Constructions of Optimal Frequency-Hopping Sequence Sets

“…< p r . If m(s) ≤ p 1 − 1, then there exists an optimal (nN, nv, λ ) FHS over Z q r × · · · × Z q 1 × F. [25] is a special case of Corollary 1 for gcd(n, N) = 1 and ω i (…”

“…In Table 2, lpf(n) denotes the least prime factor of n, q = p a for a prime p, r is a prime power for a prime α. [25] is a special case of Corollary 3 for gcd(q − 1, N) = 1 and ω i (…”

A Construction of Optimal Frequency Hopping Sequence Set via Combination of Multiplicative and Additive Groups of Finite Fields

“…Proof. By applying the (k, lpf(k) − 1; k) OC sequence set given in [2,1,8], the (p − 1, p; p) OC sequence set given in [2,1,8] and the (k(p − 1), min{lpf(k) − 1, p}; kp) OC sequence set given in [2,8], respectively to Theorem 4, we obtain the desired FHS sets in Table 1.…”