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Large families of binary sequences with low correlation values and
large linear span are critical for spread-spectrum communication
systems. The author describes a method for constructing such families
from families of homogeneous functions over finite fields, satisfying
certain properties. He then uses this general method to construct
specific fam...
Citations
... Kasami sequences can be constructed using m-sequences and their decimations [12]. Besides, there are some known families of sequences of length 2 n − 1 with good correlation properties, such as bent function sequences [25], No sequences [23], Trace Norm sequences [14]. In 2011, Zhou and Tang generalized the modified Gold sequences and obtained binary sequences with length 2 n −1 for n = 2m+1, size 2 ℓn +· · ·+2 n −1, and correlation 2 m+ℓ +1 for each 1 ℓ m [34]. ...
... Trace Norm [14] 2 n − 1, n even ...
Sequences with a low correlation have very important applications in communications, cryptography, and compressed sensing. In the literature, many efforts have been made to construct good sequences with various lengths where binary sequences attracts great attention. As a result, various constructions of good binary sequences have been proposed. However, most of the known constructions made use of the multiplicative cyclic group structure of finite field $\mathbb{F}_{p^n}$ for a prime $p$ and a positive integer $n$. In fact, all $p^n+1$ rational places including the place at infinity of the rational function field over $\mathbb{F}_{p^n}$ form a cyclic structure under an automorphism of order $p^n+1$. In this paper, we make use of this cyclic structure to provide an explicit construction of binary sequences with a low correlation of length $p^n+1$ via cyclotomic function fields over $\mathbb{F}_{p^n}$ for any odd prime $p$. Each family of binary sequences has size $p^n-2$ and its correlation is upper bounded by $4+\lfloor 2\cdot p^{n/2}\rfloor$. To the best of our knowledge, this is the first construction of binary sequences with a low correlation of length $p^n+1$ for odd prime $p$. Moreover, our sequences can be constructed explicitly and have competitive parameters.
... Kasami sequences can be constructed using m-sequences and their decimations [11]. Besides, there are some known families of sequences of length 2 n − 1 with good correlation properties, such as bent function sequences [24], No sequences [22], Trace Norm sequences [13]. In 2011, Zhou and Tang generalized the modified Gold sequences and obtained binary sequences with length 2 n − 1 for n = 2m + 1, size 2 n + · · · + 2 n − 1, and correlation 2 m+ + 1 for each 1 m [33]. ...
... For the case n ≡ 2(mod 4), compared with the Gold sequences of length 2 n − 1, our sequences have a smaller correlation and a larger length although the family size of our sequences is slightly smaller. For the case n ≡ 0(mod 4), all families of sequences given in [11], [13], [22], [24] have size 2 n/2 which is much less than the size 2 n − 1 of our family of sequences, although the correlation of their sequences is one half of our correlation. Hence, it turns out that our sequences have competitive parameters for even n. ...
Due to wide applications of binary sequences with a low correlation to communications, various constructions of such sequences have been proposed in the literature. Many efforts have been made to construct good binary sequences with various lengths. However, most of the known constructions make use of the multiplicative cyclic group structure of finite field
$\mathbb {F}_{2^{n}}$
for a positive integer
$n$
. It is often overlooked in this community that all
$2^{n}+1$
rational places (including “the place at infinity”) of the rational function field over
$\mathbb {F}_{2^{n}}$
form a cyclic structure under an automorphism of order
$2^{n}+1$
. In this paper, we make use of this cyclic structure to provide an explicit construction of binary sequences with a low correlation of length
$2^{n}+1$
via cyclotomic function fields over
$\mathbb {F}_{2^{n}}$
. Each family of our sequences has size
$2^{n}-1$
and its correlation is upper bounded by
$\lfloor 2^{(n+2)/2}\rfloor $
. To the best of our knowledge, this is the first construction of binary sequences with a low correlation of length
$2^{n}+1$
. Moreover, our sequences can be constructed explicitly and have competitive parameters. In particular, compared with the Gold sequences of length
$2^{n}-1$
for even
$n$
, our sequences have a smaller correlation and a larger length although the family size of our sequences is slightly smaller.
... Kasami sequences can be constructed using m-sequences and their decimations [9]. Besides, there are some known families of sequences of length 2 n −1 with good correlation properties, such as bent function sequences [20], No sequences [18], TN sequences [10]. In 2011, Zhou and Tang generalized the modified Gold sequences and obtained binary sequences with length 2 n − 1, n = 2m + 1, size 2 ℓn + · · · + 2 n − 1, correlation 2 m+ℓ + 1 [28]. ...
Due to wide applications of binary sequences with low correlation to communications, various constructions of such sequences have been proposed in literature. However, most of the known constructions via finite fields make use of the multiplicative cyclic group of $\F_{2^n}$. It is often overlooked in this community that all $2^n+1$ rational places (including "place at infinity") of the rational function field over $\F_{2^n}$ form a cyclic structure under an automorphism of order $2^n+1$. In this paper, we make use of this cyclic structure to provide an explicit construction of families of binary sequences of length $2^n+1$ via the finite field $\F_{2^n}$. Each family of sequences has size $2^n-1$ and its correlation is upper bounded by $\lfloor 2^{(n+2)/2}\rfloor$. Our sequences can be constructed explicitly and have competitive parameters. In particular, compared with the Gold sequences of length $2^n-1$ for even $n$, we have larger length and smaller correlation although the family size of our sequences is slightly smaller.
... As you know, d-form functions give a rich source of functions with difference balanced property, which were first defined in [26]. ...
... = (33,24,24,24,25,1,2,3,26,5,6,7,27,9,10,11,28,13,14,15,29,17,18,19,30,21,22,23,31,2,3,0,32,6,7,4,25,10,11,8,26,14,15,12,27,18,19,16,28,22,23,20,29,3,0,1,30,7,4,5,31,11,8,9,32,15,12,13,25,19,16,17,26,23,20,21,27,0,1,2,28,4,5,6,29,8,9,10,30,12,13,14,31,16,17,18,32,20,21,22). ...
... = (33,24,24,24,25,1,2,3,26,5,6,7,27,9,10,11,28,13,14,15,29,17,18,19,30,21,22,23,31,2,3,0,32,6,7,4,25,10,11,8,26,14,15,12,27,18,19,16,28,22,23,20,29,3,0,1,30,7,4,5,31,11,8,9,32,15,12,13,25,19,16,17,26,23,20,21,27,0,1,2,28,4,5,6,29,8,9,10,30,12,13,14,31,16,17,18,32,20,21,22). ...
Cyclotomy, firstly introduced by Gauss, is an important topic in Mathematics since it has a number of applications in number theory, combinatorics, coding theory and cryptography. Depending on \begin{document}$ v $\end{document} prime or composite, cyclotomy on a residue class ring \begin{document}$ {\mathbb{Z}}_{v} $\end{document} can be divided into classical cyclotomy or generalized cyclotomy. Inspired by a foregoing work of Zeng et al. [40], we introduce a generalized cyclotomy of order \begin{document}$ e $\end{document} on the ring \begin{document}$ {\rm GF}(q_1)\times {\rm GF}(q_2)\times \cdots \times {\rm GF}(q_k) $\end{document}, where \begin{document}$ q_i $\end{document} and \begin{document}$ q_j $\end{document} (\begin{document}$ i\neq j $\end{document}) may not be co-prime, which includes classical cyclotomy as a special case. Here, \begin{document}$ q_1 $\end{document}, \begin{document}$ q_2 $\end{document}, \begin{document}$ \cdots $\end{document}, \begin{document}$ q_k $\end{document} are powers of primes with an integer \begin{document}$ e|(q_i-1) $\end{document} for any \begin{document}$ 1\leq i\leq k $\end{document}. Then we obtain some basic properties of the corresponding generalized cyclotomic numbers. Furthermore, we propose three classes of partitioned difference families by means of the generalized cyclotomy above and \begin{document}$ d $\end{document}-form functions with difference balanced property. Afterwards, three families of optimal constant composition codes from these partitioned difference families are obtained, and their parameters are also summarized.
... Almost all constructions of these sequences are based on difference sets or almost difference sets. Further, d-form functions are useful tools for constructing difference sets, almost difference sets and sequences with low correlation properties [10,16,20]. Especially, d-form functions with difference-balanced properties are often associated with difference sets or sequences having ideal autocorrelation [16,17,30,31]. ...
... Let q be a power of a prime and let n be a positive integer. Klapper [10] introduced the notion of d-form function from F q n to F q and showed that sequences obtained from them have good correlation properties. ...
Pseudo-random sequences with good statistical properties, such as low autocorrelation, high linear complexity and large 2-adic complexity, have been used in designing reliable stream ciphers. In this paper, we obtain the exact autocorrelation distribution of a class of binary sequences with three-level autocorrelation and analyze the 2-adic complexity of this class of sequences. Our results show that the 2-adic complexity of such a binary sequence with period N is at least (N + 1) − log2 (N + 1). We further show that it is maximal for infinitely many cases. This indicates that the 2-adic complexity of this class of sequences is large enough to resist the attack of the rational approximation algorithm (RAA) for feedback with carry shift registers (FCSRs).
... Ces codes sont aussi des codes binaires utilisés en cryptographie. Leur génération est quasisimilaire à celle des codes de No à l'exception de l'utilisation de la fonction Norme [43]. Les caractéristiques de cette famille sont une cardinalité de √ + 1 = 2 /2 séquence de longueur avec des niveaux de corrélations (hors pic principal) bornés par √ + 1 + 1. En conclusion, malgré les problèmes relatifs au contrôle du spectre, ces trois familles semblent être de bonnes candidates pour une application MSPSR active. ...
... Les caractéristiques de cette famille sont une cardinalité de √ + 1 = 2 /2 séquence de longueur avec des niveaux de corrélations (hors pic principal) bornés par √ + 1 + 1. En conclusion, malgré les problèmes relatifs au contrôle du spectre, ces trois familles semblent être de bonnes candidates pour une application MSPSR active. TN [43] BBAG (Bruit Blanc Additif Gaussien) Toutefois, dans la partie suivante nous allons seulement caractériser les signaux OFDM ayant donnés les meilleurs résultats c'est-à-dire ceux modulés par du QPSK ou des m-séquences. [64]. ...
Aujourd’hui, les systèmes MSPSR (Multi-Static Primary Surveillance Radar) passifs se sont installés de manière durable dans le paysage de la surveillance aérienne [1]. L’intérêt que suscitent ces nouveaux systèmes provient du fait qu’en comparaison aux radars mono-statiques utilisés actuellement, les systèmes MSPSR reposent sur une distribution spatiale d’émetteurs et de récepteurs offrant des avantages en termes de fiabilité (redondance), de coûts (absence de joints tournants et émetteurs moins puissants) et de performances (diversité spatiale). Toutefois, le défaut majeur du MSPSR passif réside en l’absence de formes d’ondes dédiées due à l’exploitation d’émetteurs d’opportunités tels que les émetteurs de radio FM (Frequency Modulation) et/ou de DVB-T (Digital Video Broadcasting-Terrestrial) [2]. Afin de pallier à ce défaut, il est envisagé d’utiliser des émetteurs dédiés permettant l’emploi de formes d’ondes optimisées pour une application radar, on parle alors de MSPSR actif. Cette thèse se place dans ce cadre et a pour objectif d’étudier et de définir la ou les formes d’ondes ainsi que les traitements associés permettant d’atteindre de meilleurs performances : une meilleure flexibilité sur la disposition du système (positionnement des émetteurs libres), une continuité de service (non dépendance d’un système tiers) et de meilleurs performances radars (e.g. en terme de précision des mesures, détections, …). Dans ce but, cette thèse étudie : - Les critères de sélection des codes : comportement des fonctions d’ambiguïtés, PAPR (Peak to Average Power Ratio), efficacité spectrale, etc... ; - Les formes d’ondes utilisées en télécommunication (scrambling code, OFDM) afin d’identifier leur possible réemploi pour une application radar ; - L’utilisation d’algorithmes cycliques pour générer des familles de séquences adaptées à notre problème ; - Une approche basée sur une descente de gradient afin de générer des familles de codes de manière plus efficiente ; - Et l’évaluation des performances de ces différents algorithmes à travers l’établissement d’une borne supérieure sur le niveau maximum des lobes secondaires et à travers le dépouillement des données enregistrées suite à des campagnes d’essais
... The δ-homogeneous function from F q n to F q is introduced by Klapper [26], which is defined as H(xy) = y δ H(x) for any x ∈ F q n and y ∈ F q . Then, Kim et al. constructed an RDS from a δ-homogeneous function on F * q n [27]. ...
Fractional repetition (FR) codes are a class of distributed storage codes that replicate and distribute information data over several nodes for easy repair, as well as efficient reconstruction. In this paper, we propose three new constructions of FR codes based on relative difference sets (RDSs) with λ = 1 . Specifically, we propose new ( q 2 - 1 , q , q ) FR codes using cyclic RDS with parameters ( q + 1 , q - 1 , q , 1 ) constructed from q-ary m-sequences of period q 2 - 1 for a prime power q, ( p 2 , p , p ) FR codes using non-cyclic RDS with parameters ( p , p , p , 1 ) for an odd prime p or p = 4 and ( 4 l , 2 l , 2 l ) FR codes using non-cyclic RDS with parameters ( 2 l , 2 l , 2 l , 1 ) constructed from the Galois ring for a positive integer l. They are differentiated from the existing FR codes with respect to the constructable code parameters. It turns out that the proposed FR codes are (near) optimal for some parameters in terms of the FR capacity bound. Especially, ( 8 , 3 , 3 ) and ( 9 , 3 , 3 ) FR codes are optimal, that is, they meet the FR capacity bound for all k. To support various code parameters, we modify the proposed ( q 2 - 1 , q , q ) FR codes using decimation by a factor of the code length q 2 - 1 , which also gives us new good FR codes.
... It should be noted that all the known DB functions above are d-form functions, which are first defined in [7]. ...
By using shift sequences defined by difference balanced functions with d-form property, and column sequences defined by a mutually orthogonal almost perfect sequences pair, new almost perfect, odd perfect, and perfect sequences are obtained via interleaving method. Furthermore, the proposed perfect QAM+ sequences positively answer to the problem of the existence of perfect QAM+ sequences proposed by Boztaş and Udaya.
... Families of sequences with optimal correlation such as Kasami sequences [2], bent function sequences [3] and Gold sequences [4] have been found. Later on, other binary sequences families with low correlation are proposed, and some of them have large family size [5][6][7][8][9] or large linear span [10][11][12]. ...
... The later are specially used to design d-form sequences. Applying lifting idea to sets of d-form sequence, new families of sequences are found [10][11][12]. Some of them have large linear span [10][11][12]. ...
... Applying lifting idea to sets of d-form sequence, new families of sequences are found [10][11][12]. Some of them have large linear span [10][11][12]. Zeng [9] presented a large family by lifting the sequences which combines Niho sequences [13] and Helleseth sequences [14]. ...
In this paper, a new family S(r) of 25n/2 binary sequences of period 2n-1 is proposed, where n = 2m and gcd(r,2m -1) = 1. The presented family takes 7-valued correlation values -2n/2 -1, -1, 2n/2 -1, 2· 2n/2 -1, 3· 2n/2 -1, 4· 2n/2 -1 and 5· 2n/2 -1. For r = (2m-1 -1)/7 and m = 1 mod 3, it is proved that the proposed sequence family S(r) has linear spans n · l(n-2)/6/2, where l = 2, 3, 4, 5, 6, 7, and the distribution of linear span of sequences in S(r) is determined.
... . Then, [20]. ...
Let p be an odd prime and n=(2m+1)e. Based on the theory of quadratic forms over finite fields of odd characteristic, we generalize the binary construction by Yu and Gong to p-ary case. As a result, we obtain a new family Fok(ρ) of p-ary sequences of period pn−1 for arbitrary positive integers 1≤ρ≤m and k with gcd(n,k)=e. It is shown that, for a given ρ, Fok(ρ) has family size pnρ, maximum correlation 1+pn+(2ρ−1)e2, and maximum linear span (m+1)n. In particular, the new family Fok(ρ) contains Tang, Udaya, and Fan’s construction as a subset, if an m-sequence is excluded.
