Length Pq Research Articles

The autocorrelation of a class of quaternary sequences of length pq with high complexity

Recently, a class of quaternary sequences with period pq, where p and q are two distinct odd primes introduced by Zhang et al. were proved to possess high linear complexity and 4-adic complexity. In this paper, we determine the autocorrelation distribution of this class of quaternary sequence. Our results indicate that the studied quaternary sequence are weak with respect to the correlation property.

A wavelet-based VCG QRS loop boundaries and isoelectric coordinates detector.

This paper deals with a wavelet-based algorithm for automatic detection of isoelectric coordinates of individual QRS loops of VCG record. Fiducial time instants of QRS peak, QRS onset, QRS end, and isoelectric PQ interval are evaluated on three VCG leads together with global QRS boundaries of a record to spatiotemporal QRS loops alignment. The algorithm was developed and optimized on 161 VCG records of PTB diagnostic database of healthy control subjects (HC), patients with myocardial infarction (MI) and patients with bundle branch block (BBB) and validated on CSE multilead measurement database of 124 records of the same diagnostic groups. The QRS peak was evaluated correctly for all of 1,467 beats. QRS onset, QRS end were detected with standard deviation of 5,5ms and 7,8ms respectively from the referee annotation. The isoelectric 20ms length PQ interval window was detected correctly between the P end and QRS onset for all the cases. The proposed algorithm complies the limits for the QRS onset and QRS end detection and provides comparable or better results to other well-known algorithms. The algorithm evaluates well a wide QRS based on automated wavelet scale switching. The designed multi-lead approach QRS loop detector accomplishes diagnostic VCG processing, aligned QRS loops imaging and it is suitable for beat-to-beat variability assessment and further automatic VCG classification.

M-Ary Character-Based Sequences of Length pq With Low Autocorrelation

Using elementary properties of primitive multiplicative characters we derive the autocorrelation of a unimodular <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$M$ </tex-math></inline-formula> -ary character-based sequence of length <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$pq$ </tex-math></inline-formula> , where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula> are distinct odd primes. Subsequently: (1) we identify three new sets of twin-prime quadriphase sequences with the largest autocorrelation sidelobe magnitude equal to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\sqrt {5}$ </tex-math></inline-formula> or 3, (2) we show that the smallest maximum autocorrelation sidelobe magnitude of an <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$M$ </tex-math></inline-formula> -ary character-based sequence, when <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$4|M$ </tex-math></inline-formula> , either decreases with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$M$ </tex-math></inline-formula> and tends to 2, when <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$6\nmid M$ </tex-math></inline-formula> , or is equal to 2, when <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$6|M$ </tex-math></inline-formula> , and (3), we show that the smallest maximum autocorrelation sidelobe magnitude of a full-alphabet <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$M$ </tex-math></inline-formula> -ary character-based sequence, where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$M&gt;2$ </tex-math></inline-formula> , is greater or equal to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\sqrt {3}$ </tex-math></inline-formula> , and, in particular, is equal to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\sqrt {3}$ </tex-math></inline-formula> for the sextic phase sequence.

Cyclic self dual codes of length pq over Z 4

Explicit expressions for primitive idempotent in the ring are obtained, where p and q are distinct odd primes with multiplicative order of 2 modulo p and modulo q being and ϕ(q) respectively. Hence the idempotent generators of cyclic self dual codes of length pq over Z 4 are obtained. Further, it is observed that when p = 8k - 1 and q = 8m - 3, the extension of each of these self dual codes augmented with all-ones vector is a Type I code.

Quaternary Sequence of Length pq with Low Autocorrelation

Using inverse Gray mapping and generalized cyclotomic binary sequences, Zheng Yang et al.[2] constructed a quaternary sequence of length pq with low autocorrelation. In this paper, we present a new quaternary sequence by a variant of inverse Gray mapping and the generalized cyclotomic binary sequences, and calculate autocorrelation function of the new quaternary sequence. The results show that these two quaternary sequences have the same maximum nontrivial autocorrelation magnitude. Therefore it provides more quaternary sequences with low autocorrelation in communication systems.

Linear complexity of a class of pseudorandom sequences over a general finite field

Due to their nice algebraic structures and pseudorandom features, generalized cyclotomic sequences have wide applications in simulation, coding and cryptography. Based on the Ding–Helleseth sequence, Bai et al. proposed a class of balanced generalized sequences of length pq. Moreover, they showed that this class of sequences has high linear complexity over a finite field of order two. In this paper, we study the linear complexity and the minimal polynomial of this class of sequences over a general finite field. Results indicate the sequence considered possesses high linear complexity over a general finite field.

Notes about the linear complexity of Ding-Helleseth generalized cyclotomic sequences of lengthpqover the finite field of orderporq

We investigate three classes of Ding-Helleseth-generalized cyclotomic sequences of length pq . We derive the linear complexity and the minimal polynomial of above-mentioned sequences over the finite fields of orders p and q , where p and q are two odd distinct primes, and obtain series of sequences with high linear complexity.

On the k-error linear complexity of generalised cyclotomic sequences

Generalised cyclotomic binary sequences are divided into Whiteman-generalised cyclotomic and Ding-generalised cyclotomic. Firstly, two classes of error generalised cyclotomic sequences with length pq over Z

On the k-error linear complexity of generalised cyclotomic sequences

Generalised cyclotomic binary sequences are divided into Whiteman-generalised cyclotomic and Ding-generalised cyclotomic. Firstly, two classes of error generalised cyclotomic sequences with length pq over Zpq are constructed; secondly, k-error linear complexity of generalised cyclotomic sequences are considered, using the trace representation and cyclotomic theory, respectively. It is shown that k-error linear complexity of generalised cyclotomic sequences do not exceed p + q-1 which is much less than the (zero-error) linear complexity.

Linear complexity of quaternary sequences of length [formula omitted] with low autocorrelation

We derive the linear complexity of quaternary sequences of length pq with low autocorrelation over the finite field of four elements and over the finite ring of order 4. Also we examine the linear complexity of the other sequences of length pq.

Pseudo-Randomness of Certain Sequences of k Symbols with Length pq

Pseudo-Randomness of Certain Sequences of k Symbols with Length pq

On Generalized Cyclotomic Sequence of Order d and Period pq

In this letter, we generalize the binary sequence introduced by Li et al. in [S. Q. Li et al., On the randomness generalized cyclotomic sequences of order two and length pq, IEICE Trans. Fund, vol. E90-A, no.9, pp.2037-2041, 2007] to sequence over arbitrary prime fields. Furthermore, the auto-correlation distribution and linear complexity of the proposed sequence are presented.

Linear complexity and autocorrelation values of a polyphase generalized cyclotomic sequence of length pq

A construction of a family of generalized polyphase cyclotomic sequences of length pq is presented in terms of the generalized cyclotomic classes modulo pq. Their linear complexity and corresponding minimal polynomials are deduced. Some upper bounds on periodic and aperiodic autocorrelation values of resulting sequences are also estimated by using certain exponential sums.

Construction of quaternary sequences of length pq with low autocorrelation

In this paper, a new quaternary sequences over Z pq of length pq is constructed by using inverse Gray mapping and generalized cyclotomic sequences over Z p and Z q . The maximum nontrivial autocorrelation of the constructed sequences are shown to be p???q?+?3 or $\max\{q-p-1, \sqrt{5}\}$ .

Generalized M-Ary Related-Prime Sequences with Low Correlation

In this paper we introduce new M-ary sequences of length pq, called generalized M-ary related-prime sequences, where p and q are distinct odd primes, and M is a common divisor of p - 1 and q - 1. We show that their out-of-phase autocorrelation values are upper bounded by the maximum between q - p + 1 and 5. We also construct a family of generalized M-ary related-prime sequences and show that the maximum correlation of the proposed sequence family is upper bounded by p + q - 1.

Some Notes on Generalized Cyclotomic Sequences of Length pq

We review the constructions of two main kinds of generalized cyclotomic binary sequences with length pq (the product with two distinct primes). One is the White-generalized cyclotomic sequences, the other is the Ding-Helleseth(DH, for short)-generalized cyclotomic sequences. We present some new pseudo-random properties of DH-generalized cyclotomic sequences using the theory of character sums instead of the theory of cyclotomy, which is a conventional method for investigating generalized cyclotomic sequences.

Trace representation of some generalized cyclotomic sequences of length pq

This paper contributes to trace representation of some generalized cyclotomic sequences of length pq ( p , q prime ) , which are defined by Ding and Helleseth. From the relations between these sequences and the Legendre sequence, we firstly confirm the defining pairs of these sequences of arbitrary order. Then, we obtain their trace representation, from which we give their linear complexity using Key’s method. It can be seen that Bai et al.’s conclusion is a special case of our result when the order is two. Finally, an example is given to illustrate the validity of our result.

Autocorrelation Values of New Generalized Cyclotomic Sequences of Order Two and Length pq

Pseudo-random sequences are used extensively for their high speed and security level and less errors. As a branch, the cyclotomic sequences and the generalized ones are studied widely because of their simple mathematical structures and excellent pseudo-random properties. In 1998, Ding and Helleseth introduced a new generalized cyclotomy which includes the classical cyclotomy as a special case. In this paper, based on the generalized cyclotomy, new generalized cyclotomic sequences with order two and length pq are constructed. An equivalent definition of the sequences is deduced so that the autocorrelation values of these sequences can be determined conveniently. The construction contributes to the understanding of the periodic autocorrelation structure of cyclotomically-constructed binary sequences, and the autocorrelation function takes on only a few values.

On the Randomness of Generalized Cyclotomic Sequences of Order Two and Length pq

In this letter we introduce new generalized cyclotomic sequences of order two and length pq firstly, then we determine the linear complexity and autocorrelation values of these sequences. Our results show that these sequences are rather good from the linear complexity viewpoint.

Classification of some metric automorphisms defined by Standish

Let F be a finite set with a probability distribution { P i : i ϵ F} and (Ω F , P) denote the product space of countably many copies of ( F, P). A permutation (bijection) φ of the integers induces an invertible measure preserving transformation T φ on (Ω F , P) given by the equation ( T φ w) i = w φ( j) . Such metric automorphisms we call S-automorphisms. We show in this paper that S-automorphisms are ergodic if and only they are Bernoulli shifts and two ergodic S-automorphisms are isomorphic if and only if their associated permutations are conjugate. We also show that S-automorphisms have discrete spectrum if and only if they have zero entropy and every S-automorphism is either a Bernoulli shift, has discrete spectrum, or is a product of a Bernoulli shift and an automorphism with discrete spectrum. S-automorphism with discrete spectrum are those whose associated permutations contain only cycles of finite length. These automorphisms are studied according to the number of such finite cycles. Those whose permutations have infinitely many finite cycles with unbounded lengths are shown to be antiperiodic and those whose permutations have infinitely many finite cycles of bounded length are periodic with almost no fixed points. An example is given of two automorphisms of this latter type which are isomorphic but whose permutations are not conjugate. A complete isomorphism invariant is given for S-automorphisms whose associated permutations consist of finitely many finite cycles. Using this invariant we show that if φ is either a product of k disjoint cycles of prime power p α , or a single cycle of length pq where p and q are primes, or a product of k disjoint cycles of prime lengths p 1 < p 2 < ··· < p k and if ψ is a permutation such that T ψ and T φ are isomorphic then ψ is conjugate to φ.