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In this paper, we propose the zero-correlation-zone (ZCZ) of radius r on two-dimensional m × n sonar sequences and define the (m , n , r) ZCZ sonar sequences. We also define some new optimality of an (m , n , r) ZCZ sonar sequence which has the largest r for given m and n. Because of the ZCZ for perfect autocorrelation, we are able to relax the distinct difference property of the conventional sonar sequences, and hence, the autocorrelation of ZCZ sonar sequences outside ZCZ may not be upper bounded by 1. We may sometimes require such an ideal autocorrelation outside ZCZ, and we define ZCZ-DD sonar sequences, indicating that it has an additional distinct difference (DD) property. We first derive an upper bound on the ZCZ radius r in terms of m and n ≥ m . We next propose some constructions for (m , n , r) ZCZ sonar sequences, which leads to some very good constructive lower bound on r. Furthermore, this construction suggests that for m and r, the parameter n can be as large as possible indefinitely. We present some exhaustive search results on the existence of (m , n , r) ZCZ sonar sequences for some small values of r. For ZCZ-DD sonar sequences, we prove that some variations of Costas arrays construct some ZCZ-DD sonar sequences with ZCZ radius r = 2 . We also provide some exhaustive search results on the existence of (m , n , r) ZCZ-DD sonar sequences. Lots of open problems are listed at the end. [ABSTRACT FROM AUTHOR]

This paper proposes a modification to Tanner's work for constructing girth‐8 quasi‐cyclic low‐density parity‐check codes. The main contribution of this paper is to use an arithmetic sequence at the leftmost column for the exponent matrix so that the lifting size is not necessarily restricted to the prime numbers. Two theorems on the lifting sizes that achieve girth at least 8 using this approach is also provided. This construction exhibits better frame error rate results to the modified 5G new radio (NR) low‐density parity‐check codes for lengths around 500. Also, this construction achieves better frame error rate performance results than the recently proposed one using the Golomb rulers at around frame error rate of 10−6. [ABSTRACT FROM AUTHOR]

Sidelnikov sequences, a kind of cyclotomic sequences with many desired properties such as low correlation and variable alphabet sizes, can be employed to construct a polyphase sequence family that has many applications in high-speed data communications. Recently, cyclotomic numbers have been used to investigate the linear complexity of Sidelnikov sequences, mainly about binary ones, although the limitation on the orders of the available cyclotomic numbers makes it difficult. This paper continues to study the linear complexity over $\mathbb {F}_{M}$ of $M$ -ary Sidelnikov sequence of period $q-1$ using Hasse derivative, which implies $q=p^{m}$ , $m\geq 1$ and $M|(q-1)$. The $t$ th Hasse derivative formulas are presented in terms of cyclotomic numbers, and some upper bounds on the linear complexity for $M=2$ and 3 are obtained only with some additional restrictions on $q$. Furthermore, concrete illustrations for several families of these sequences, such as $q\equiv 1\pmod {2}$ and $q\equiv 1\pmod {3}$ , show these upper bounds are tight and reachable; especially for $q=2\times 3^{\lambda }+1 (1\leq \lambda \leq 20)$ , the exact linear complexities over $\mathbb {F}_{3}$ of the ternary Sidelnikov sequences are determined; and it turns out that all the linear complexities of the sequences considered are very close to their periods. [ABSTRACT FROM AUTHOR]

In this correspondence, we present a connection between designing low-correlation zone (LCZ) sequences and the results of correlation of sequences with subfield decompositions presented in a recent book by the first two authors. This results in LCZ signal sets with huge sizes over three different alphabetic sets: finite field of size q, integer residue ring modulo q, and the subset in the complex field which consists of powers of a primitive q th root of unity. We show a connection between these sequence designs and "completely noncyclic" Hadamard matrices and a construction for those sequences. We also provide some open problems along this direction. Index Terms--Hadamard matrices of completely noncyclic type, low-correlation zone (LCZ) sequences, subfield reducible sequences, two-tuple balance property.

In this correspondence, we prove that the complex crosscorrelation of a k-ary Sidel'nikov sequence of period q - 1 and its constant multiple sequence is upper bounded by [square root of (q)]/+ 3, where q = [p.sup.m] and here p is an odd prime and m is a positive integer. Index Terms--Autocorrelation, cross correlation, Sidel'nikov sequences.

We classify some [p.sup.k]-ary (p prime, k integer) generalized m-sequences and generalized Gordon-Mills--Welch (GMW) sequences of period [p.sup.2k]--1 over a residue class ring R = GF(p)[xi][??]/([[xi].sup.k]) having optimal partial Hamming autocorrelation properties. In frequency hopping (FH) spread-spectrum systems, these sequences are useful for synchronizing process. Suppose, for example, that a transmitting [p.sup.k]-ary FH patterns of period [p.sup.2k] - 1 are correlated at a receiver. Usually, the length of a correlation window, denoted by L, is shorter than the pattern's overall period. In that case, the maximum value of the out-of-phase Hamming autocorrelation is lower-bounded by [L/[p.sup.k] + 1] but the classified sequences achieve this bound with equality for any positive integer L. Index Terms--Finite rings, frequency hopping, generalized Gordon-Mills-Welch (GGMW) sequences, Hamming correlation, partial autocorrelation.