Generalized quasi-cyclic (GQC) codes with arbitrary lengths over the ring $mathbb{F}_{q}+umathbb{F}_{q}$, where u2=0, q=pn, n a positive integer and p a prime number, are investigated. By the Chinese Remainder Theorem, structural properties and the decomposition of GQC codes are given. For 1-generator GQC codes, minimum generating sets and lower bounds on the minimum distance are given.

The average Hamming correlation is an important performance indicator of frequency-hopping sequences (FHSs). In this letter, the average partial Hamming correlation (APHC) properties of FHSs are discussed. Firstly, the theoretical bound on the average partial Hamming correlation of FHSs is established. It works for any correlation window with length 1≤ω≤υ, where υ is the sequence period, and generalizes the bound developed by Peng et al which is valid only when ω=υ. A sufficient and necessary condition for a set of FHSs having optimal APHC for any correlation window is then given. Finally, sets of FHSs with optimal APHC are presented.

In this letter, we first investigate some new properties of a known power residue frequency-hopping sequence (FHS) set which is established as an optimal one-coincidence frequency-hopping sequence (OC-FHS) set with near-optimal set size. Next, combining the mathematical structure of power residue theory with interleaving technique, we present a new class of optimal OC-FHS set, using the Chinese Remainder Theorem (CRT). As a result, one optimal OC-FHS set with prime length is extended to another optimal OC-FHS set with composite length in which the construction preserves the maximum Hamming correlation (MHC) and the set size as well as the optimality of the Lempel-Greenberger bound.