The paper offers results concerning partial periodic correlations of pseudo-random sequences, including classical results on binary $m$-sequences but also recent results on the first two partial period correlation moments of sequences in families $A$, $B$ and $C$ over Galois rings. Traditionally, sequence families were designed using Galois field theory. More recently, Galois rings have been used for the design of CDMA (Code Division Multiple Access) sequence families, offering better system performance. The current paper presents results obtained in the Galois ring setting, focusing on a notion which gives a significant contribution to the performance of systems: the notion of partial period correlation of sequences. The paper comprises four main sections, except for introduction and conclusions. Section 3 provides a brief overview on the structure of Galois rings, defines the sequence families $A$, $B$ and $C$ and related Cayley tables. In Section 4, the first moment of the partial correlation for sequences in family $A$ is defined and determined, and also the full period autocorrelation functions for the families $B$ and $C$. In Section 5, the second moment (local and global) of the partial correlation function for family $A$ is computed. A special remark on Section 2: it contains notions and results defined once more later on (Definition 2.2 is the same as Definition 4.2, just as Theorem 2.1 and Theorem 4.1, Theorem 4.2 and Theorem 2.1, Proposition 2.1 and Proposition 5.1, etc.). Moreover, many notations are used without definition or are wrongly formulated (e.g., Lemma 2.1). The paper is clearer without this section and it is strange that it was accepted to be published in this form. Further problems appear also in other sections (for example, on page 15 some references are directed to assertions that do not exist: Lemma 3, Theorem 4). From this point of view, the paper must be seriously improved if the authors wish to publish it in a journal. But, regardless of these weaknesses (and ignoring Section 2), we note that the present work extends significantly the results obtained in previous works of the same authors [“Partial correlations of Galois field sequences”, Proc. IEEE IWSDA Workshop, 157‒161, Chengdu, China, (2007); “On partial correlations of various $Z_4$ sequence families”, Lect. Notes Comput. Sci. 5203, 332‒344 (2008; Zbl 1206.94045)].
Reviewer: Adrian Atanasiu (Bucharest)