Nonbinary Sequences Research Articles

Sets of Nonbinary Sequences with a Low Level of Mutual Correlation for Systems of Digital Information Transmission

The sets of nonbinary pseudorandom sequences (NPSs) for periods N = p^S– 1 20 000 (p = 3, 5,7, and 11) generated in finite fields GF(pS) whose power is V = N + 1 and the maximum of the module of peaks of the periodic autocorrelation function (PACF) and periodic cross-correlation function (PCCF) satisfy the bounds obtained by Sidel’nikov. In addition to the minimal polynomials of elements α and α^2 , where α is a primitive element of field GF(p^S), the minimal polynomials of elements α and α^(i_d)(i_d is the decimation index) are determined on the basis of which the new sets of NPSs can be formed with the equivalent correlation properties. Sets of indices i_d 2 are determined for various combinations of parameters p and S. The cases of even and odd values of parameter S are considered, for which the values of the PACF and PCCF with the maximum module are obtained and the number and values of different levels of correlation functions are determined.

Sets of Nonbinary Sequences with a Low Level of Mutual Correlation for Systems of Digital Information Transmission

Sets of Nonbinary Sequences with a Low Level of Mutual Correlation for Systems of Digital Information Transmission

Sequential universal modeling for non-binary sequences with constrained distributions

Sequential probability assignment and universal compression go hand in hand. We propose sequential probability assignment for non-binary (and large alphabet) sequences with empirical distributions whose parameters are known to be bounded within a limited interval. Sequential probability assignment algorithms are essential in many applications that require fast and accurate estimation of the maximizing sequence probability. These applications include learning, regression, channel estimation and decoding, prediction, and universal compression. On the other hand, constrained distributions introduce interesting theoretical twists that must be overcome in order to present efficient sequential algorithms. Here, we focus on universal compression for memoryless sources, and present precise analysis for the maximal minimax and the average minimax for constrained distributions. We show that our sequential algorithm based on modified Krichevsky-Trofimov (KT) estimator is asymptotically optimal up to $O(1)$ for both maximal and average redundancies. This paper follows and addresses the challenge presented in \cite{stw08} that suggested results for the binary case lay the foundation to studying larger alphabets.

Formation of Quinary Gordon-Mills-Welch Sequences for Discrete Information Transmission Systems

An algorithm for the formation of the quinary Gordon-Mills-Welch sequences (GMWS) with a period of N=54-1=624 over a finite field with a double extension GF[(52)2] is proposed. The algorithm is based on a matrix representation of a basic M-sequence (MS) with a primitive verification polynomial hмs(x) and a similar period. The transition to non-binary sequences is determined by the increased requirements for the information content of the information transfer processes, the speed of transmission through communication channels and the structural secrecy of the transmitted messages. It is demonstrated that the verification polynomial hG(x) of the GMWS can be represented as a product of fourth-degree polynomials-factors that are indivisible over a simple field GF(5). The relations between roots of the polynomial hмs(x) of the basic MS and roots of the polynomials hсi(x) are obtained. The entire list of GMWS with a period N=624 can be formed on the basis of the obtained ratios. It is demonstrated that for each of the 48 primitive fourth-degree polynomials that are test polynomials for basis MS, three GMWS with equivalent linear complexity (ELC) of ls=12, 24, 40 can be formed. The total number of quinary GMWS with period of N=624 is equal to 144. A device for the formation of a GMWS as a set of shift registers with linear feedbacks is presented. The mod5 multipliers and summators in registers are arranged in accordance with the coefficients of indivisible polynomials hсi(x). The symbols from the registers come to the adder mod5, on the output of which the GMWS is formed. Depending on the required ELC, the GMWS forming device consists of three, six or ten registers. The initial state of cells of the shift registers is determined by the decimation of the symbols of the basic MS at the indexes of decimation, equal to the minimum of the exponents of the roots of polynomials hсi(x). A feature of determining the initial States of the devices for the formation of quinary GMWS with respect to binary sequences is the presence of cyclic shifts of the summed sequences by a multiple of N/(p–1). The obtained results allow to synthesize the devices for the formation of a complete list of 144 quinary GMWS with a period of N=624 and different ELC. The results can also be used to construct other classes of pseudo-random sequences that allow analytical representation in finite fields.

A Construction for Balancing Non-Binary Sequences Based on Gray Code Prefixes

We introduce a new construction for the balancing of non-binary sequences that make use of Gray codes for prefix coding. Our construction provides full encoding and decoding of sequences, including the prefix. This construction is based on a generalization of Knuth’s parallel balancing approach, which can handle very long information sequences. However, the overall sequence—composed of the information sequence, together with the prefix—must be balanced. This is reminiscent of Knuth’s serial algorithm. The encoding of our construction does not make use of lookup tables, while the decoding process is simple and can be done in parallel.

Generalized nonbinary sequences with perfect autocorrelation, flexible alphabets and new periods

We extend the parameters and generalize existing constructions of perfect autocorrelation sequences over complex alphabets. In particular, we address the PSK+ constellation (Boztas and Udaya 10) and present an extended number theoretic criterion which is sufficient for the existence of the new sequences with perfect autocorrelation. These sequences are shown to exist for nonprime alphabets and more general lengths in comparison to existing designs. The new perfect autocorrelation sequences provide novel alternatives for wireless communications and radar system designers for applications in ranging and synchronisation as well as channel identification.

Exact Reconstruction From Insertions in Synchronization Codes

This paper studies problems in data reconstruction, an important area with numerous applications. In particular, we examine the reconstruction of binary and nonbinary sequences from synchronization (insertion/deletion-correcting) codes. These sequences have been corrupted by a fixed number of symbol insertions (larger than the minimum edit distance of the code), yielding a number of distinct traces to be used for reconstruction. We wish to know the minimum number of traces needed for exact reconstruction. This is a general version of a problem tackled by Levenshtein for uncoded sequences. We introduce an exact formula for the maximum number of common supersequences shared by sequences at a certain edit distance, yielding an upper bound on the number of distinct traces necessary to guarantee exact reconstruction. Without specific knowledge of the code words, this upper bound is tight. We apply our results to the famous single deletion/insertion-correcting Varshamov–Tenengolts (VT) codes and show that a significant number of VT code word pairs achieve the worst case number of outputs needed for exact reconstruction. We also consider extensions to other channels, such as adversarial deletion and insertion/deletion channels and probabilistic channels.

Deterministic Randomness Extraction from Generalized and Distributed Santha--Vazirani Sources

A Santha--Vazirani (SV) source is a sequence of random bits where the conditional distribution of each bit, given the previous bits, can be partially controlled by an adversary. Santha and Vazirani show that deterministic randomness extraction from these sources is impossible. In this paper, we study the generalization of SV sources for nonbinary sequences. We show that unlike the binary setup of Santha and Vazirani, deterministic randomness extraction in the generalized case is sometimes possible. In particular, if the adversary has access to $s$ “nondegenerate” dice that are $c$-sided and can choose one die to throw based on the previous realizations of the dice, then deterministic randomness extraction is possible if $s<c$. We present a necessary condition and a sufficient condition for the possibility of deterministic randomness extraction. These two conditions complement each other in the nondegenerate cases. Next, we turn to a distributed setting. In this setting the SV source consists of a random sequence of pairs $(a_1, b_1), (a_2, b_2), \ldots$ distributed between two parties, where the first party receives $a_i$'s and the second one receives $b_i$'s. The goal of the two parties is to extract common randomness without communication. Using the notion of maximal correlation, we prove a necessary condition and a sufficient condition for the possibility of common randomness extraction from these sources. Based on these two conditions, the problem of common randomness extraction essentially reduces to the problem of randomness extraction from (nondistributed) SV sources. This result generalizes results of Gács and Körner, and Witsenhausen about common randomness extraction from independently and identically distributed sources to adversarial sources.

A new nonbinary sequence family with low correlation and large size

Let $p$ be an odd prime, $n≥q3$ and $k$ positive integers with $e=\gcd(n,k)$ . In this paper, a new family $\mathcal{S}$ of $p$ -ary sequences with period $N=p^n-1$ is proposed. The sequences in $\mathcal{S}$ are constructed by adding a $p$ -ary sequence to its two decimated sequences with different phase shifts. The correlation distribution among sequences in $\mathcal{S}$ is completely determined. It is shown that the maximum magnitude of nontrivial correlations of $\mathcal{S}$ is upper bounded by $p^e\sqrt{N+1}+1$ , and the family size of $\mathcal{S}$ is $N^2$ . Our sequence family has a large family size and low correlation.

New evolutionary search for long low autocorrelation binary sequences

Binary sequences with low aperiodic autocorrelation levels, defined in terms of the peak sidelobe level (PSL) and/or merit factor, have many important engineering applications, such as radars, sonars, spread spectrum communications, system identification, and cryptography. Searching for low autocorrelation binary sequences (LABS) is a notorious combinatorial problem, and has been chosen to form a benchmark test for constraint solvers. Due to its prohibitively high complexity, an exhaustive search solution is impractical, except for relatively short lengths. Many suboptimal algorithms have been introduced to extend the LABS search for lengths of up to a few hundred. In this paper, we address the challenge of discovering even longer LABS by proposing an evolutionary algorithm (EA) with a new combination of several features, borrowed from genetic algorithms, evolutionary strategies (ES), and memetic algorithms. The proposed algorithm can efficiently discover long LABS of lengths up to several thousand. Record-breaking minimum peak sidelobe results of many lengths up to 4096 have been tabulated for benchmarking purposes. In addition, our algorithm design can be easily adapted to tackle various extensions of the LABS problem, say, with a generic sidelobe criterion and/or for possibly nonbinary sequences.

Noise‐based spreading in code division multiple access systems for secure communications

SUMMARYBinary spreading sequences or chaotic nonbinary sequences are used in code division multiple access systems. In contrast to those sequences, this paper presents the theory, simulation, and implementation in digital signal processor technology of a code division multiple access system that uses random spreading sequences. The expressions for probability of bit error are derived for the cases when the additive white Gaussian noise and fading are present in the channel. Because the overall transmitted signal is a sum of random signals, the security of signal transmission in this communication system can be enormously increased and will depend on the goodness of the random sequence generator. The developed theory was confirmed by a prototype of the system that was designed in digital signal processor technology. The prototype used spreading sequences that had statistical characteristics very close to the characteristics of theoretical random sequences. The main problem in the system is how to synchronize the received spread sequence with the locally generated sequence. Hence, a mathematical model of the synchronization block for the system has been separately designed and developed. The synchronization procedure has been demonstrated on the basis of application of a periodically repeated pilot sequence. Copyright © 2014 John Wiley &amp; Sons, Ltd.

Fast Time-Recursive Block Correlators for Pseudorandom Sequences

Correlators for pseudorandom sequences are used in direct sequence spread spectrum communication systems for signal synchronization and identification of transmitters. Implementation aspects of these correlators are critical for real-time processing of signals. This paper presents fast correlator structures in time domain which significantly reduce redundant operations using time-recursive and block processing. The approach can be extended to other applications which use correlations with both binary and non-binary sequences.

Good Synchronization Sequences for Permutation Codes

For communication schemes employing Frequency Hopping/Multiple Frequency Shift Keying modulation, we present an algorithm for finding good non-binary synchronization sequences, which are permutations, to be used with permutation codes to synchronize/resynchronize data in channels with background noise and interference(frequency jamming/fading). For the synchronization sequences, new analytical expressions for the probability of false acquisition are also given. Using simulation results, we show that our synchronization sequences perform better than some conventional non-binary synchronization sequences, in the presence of background noise and interference.

New nonbinary sequence families with low correlation, large size, and large linear span

Let p be an odd prime and n = ( 2 m + 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 F o k ( ρ ) of p -ary sequences of period p n − 1 for arbitrary positive integers 1 ≤ ρ ≤ m and k with gcd ( n , k ) = e . It is shown that, for a given ρ , F o k ( ρ ) has family size p n ρ , maximum correlation 1 + p n + ( 2 ρ − 1 ) e 2 , and maximum linear span ( m + 1 ) n . In particular, the new family F o k ( ρ ) contains Tang, Udaya, and Fan’s construction as a subset, if an m -sequence is excluded.

Random Binary and Non-Binary Sequences Derived from Random Sequence of Points on Cyclic Elliptic Curve Over Finite Field GF(2 m ) and Their Properties

In this paper, Cyclic Elliptic Curves of the form y 2 + xy = x 3 + ax 2 + b,a,b ∈ GF(2 m ) with order M is considered. A finite field GF(p) (p ≥ N, where N is the order of point P) is considered. Random sequence {k i } of integers is generated using Linear Feedback Shift Register (LFSR) over GF(p) for maximum period. Every element in sequence {k i } is mapped to k i P which is a point on Cyclic Elliptic Curve with co-ordinates say (x i , y i ). The sequence {k i P} is a random sequence of elliptic curve points. From the sequence (x i , y i ) several binary and non-binary sequences are derived and their randomness properties are investigated. The results are discussed. It is found that these sequences pass FIPS-140, NIST tests and exhibit good Hamming Correlation properties. These sequences find applications in Stream Cipher Systems. Here, Cyclic Elliptic Curve over GF(28) is chosen for analysis.

Families of Sequence Pairs with Zero Correlation Zone

A family of sequences with zero correlation zone, which is shortly called a ZCZ set, can provide CDMA system without co-channel interference nor influence of multipath. This paper presents two types of ZCZ sets of non-binary sequence pairs, which achieve the upper bound of family size for length and zero correlation zone. One, which is produced by use of a perfect complementary pair and an orthogonal code, can change zero correlation zone, while the upper bound is kept. The other, which is generated by use of a newly defined orthogonal pair and an orthogonal code, can offer such CDMA system as a binary ZCZ set seems to be used.

Two-tuple balance of non-binary sequences with ideal two-level autocorrelation

Let p be a prime, q = p m and F q be the finite field with q elements. In this paper, we will consider q-ary sequences of period q n - 1 for q > 2 and study their various balance properties: symbol-balance, difference-balance, and two-tuple-balance properties. The array structure of the sequences is introduced, and various implications between these balance properties and the array structure are proved. Specifically, we prove that if a q-ary sequence of period q n - 1 is difference-balanced and has the “cyclic” array structure then it is two-tuple-balanced. We conjecture that a difference-balanced q-ary sequence of period q n - 1 must have the cyclic array structure. The conjecture is confirmed with respect to all of the known q-ary sequences which are difference-balanced, in particular, which have the ideal two-level autocorrelation function when q = p .

A Generalization of Niho’s Theorem

In this note, we generalize Niho's theorem on cross-correlation functions of binary m-sequences. Our theorem applies both to binary and non-binary sequences.

A New Family of Nonbinary Sequences With Three-Level Correlation Property and Large Linear Span

In this correspondence, we present a new family of nonbinary sequences with three-level nontrivial correlations and large linear complexity. The sequences may be considered as nonlinear analogues of the well-known sequences by Trachtenberg and Helleseth. It is shown that the family is optimal with respect to the Welch bound in terms of root mean square of all nontrivial correlations. We also determine the correlation distribution of the new family.

Spherical sequences with low aperiodic crosscorrelation

The authors present a new procedure for the generation of non-binary sequences having superior aperiodic crosscorrelation properties to Gold sequences. The proposed procedure randomly extracts points from the surface of a multidimensional sphere, with a posterior selection technique similar to those used in genetic algorithms. A theoretical model is provided, describing the behaviour of these sequences, and several numerical experiments are performed comparing these sequences with those from the literature. It is demonstrated that these sequences outperform both Gold and chaotic generated sequences and approximate the performance of 4-phase sequences in aperiodic crosscorrelation. The sequences have the additional advantages of inherent privacy, due to their random nature and the unlimited number of sequences per set.