Ary Sequences Research Articles

An Efficient Algorithm for Computing the k-Error Linear Complexity Spectrum of Periodic Sequences

An efficient algorithm for computing the k-error linear complexity spectrum of a q- ary sequence s with period 2 pn is presented, where q is an odd prime and a primitive root modulo p2. The algorithm generalizes both the Wei-Xiao-Chen and the Wei algorithms, The new algorithm can compute the k-error linear complexity spectrum of s using at most 4 n+1 steps.

A Three-Valued Walsh Transform From Decimations of Helleseth–Gong Sequences

The Walsh transform of two-level autocorrelation sequences has played an important role in the construction of the set of sequences in which any two sequences are orthogonal. For p -ary sequences, there are only two basic classes of two-level autocorrelation sequences with no subfield structures for an arbitrary odd prime p. One is the class of p-ary m-sequences and the other is the class of p -ary Helleseth-Gong sequences. In this paper, the Walsh transform of a subclass of p -ary Helleseth-Gong sequences and the Walsh transform of their particular decimations are completely determined and are shown to be three-valued.

Analysis of Cross-Correlation of m-sequences and Equation on Finite Fields

ary sequences of period     are widely used in many areas of engineering and sciences. Some well-known applications include coding theory, code-division multiple-access (CDMA) communications, and stream cipher systems. The analysis of cross-correlations of these sequences is a very important problem in -ary sequences research. In this paper, we analyze cross-correlations of -ary sequences which is associated with the equation      over finite fields.

Linear complexity of pseudorandom sequences generated by Fermat quotients and their generalizations

We use polynomial quotients modulo an odd prime p, which are generalized from the Fermat quotients, to define two families of d ( ⩾ 2 ) -ary sequences of period p 2 . If d is a primitive element modulo p 2 , we determine the minimal characteristic polynomials of the sequences and hence their linear complexities, which depend on whether p ≡ 1 or 3 (mod 4). Moreover, we generalize the result to the polynomial quotients modulo a power of p.

A Generalization of the Family of $p$-ary Decimated Sequences With Low Correlation

Let p be a prime and n a positive integer. Let e|pn-1 and N=pn-1/e . In this paper, we construct a family S of e2N p -ary sequences, each member of S has period N and the magnitudes of correlations of members of S are upper bounded by 2√(pn)=2√(eN+1).

A New Family of $p$-Ary Sequences of Period $(p^n-1)/2$ With Low Correlation

For an odd prime p congruent to 3 modulo 4 and an odd integer n, a new family of p-ary sequences of period N=(pn-1)/2 with low correlation is proposed. The family is constructed by shifts and additions of two decimated m-sequences with the decimation factors 2 and 2d, d = N - pn-1. The upper bound for the maximum magnitude of nontrivial correlations of this family is derived using well known Kloosterman sums. The upper bound is shown to be 2√(N+1/2) = √(2pn) , which is twice the Welch's lower bound and approximately 1.5 times the Sidelnikov's lower bound. The size of the family is 2(pn-1) , which is four times the period of sequences.

A Construction of a New Family of $M$-ary Sequences With Low Correlation From Sidel'nikov Sequences

In this paper, a new family of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</i> -ary sequences of period <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> -1 is proposed. The proposed family is constructed by the addition of cyclic shifts of an <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</i> -ary Sidel'nikov sequence and its reverse sequence. The number of sequences contained in this family is about ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</i> -1) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> times of their period and the maximum magnitude of their correlation values is upper bounded by 4 √( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">)</i> +5.

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.

New Construction of $M$-Ary Sequence Families With Low Correlation From the Structure of Sidelnikov Sequences

For prime p and a positive integer m , it is shown that M-ary Sidelnikov sequences of period p2m-1, if M | pm-1, can be equivalently generated by the operation of elements in a finite field GF(pm), including a pm-ary m -sequence. From the (pm-1) ×(pm+1) array structure of the sequences, it is then found that a half of the column sequences and their constant multiples have low correlation enough to construct new M -ary sequence families of period pm-1. In particular, new M-ary sequence families of period pm-1 are constructed from the combination of the column sequence families and known Sidelnikov-based sequence families, where the new families have larger family sizes than the known ones with the same maximum correlation magnitudes. Finally, it is shown that the new M -ary sequence family of period pm-1 and the maximum correlation magnitude 2√{pm}+6 asymptotically achieves √2 times the equality of the Sidelnikov's lower bound when M=pm-1 for odd prime p.

On the Cross-Correlation Distributions of $M$-ary Multiplicative Character Sequences

It is well known that the magnitude of the cross correlation between any distinct constant multiple sequences of an M-ary power residue sequence of period p is upper bounded by radicp +2 and that of an M -ary Sidel'nikov sequence of period p m-1 is upper bounded by radic{p m} +3, where p is a prime and m is a positive integer. In this paper, we first show that their cross-correlation functions are closely related to Jacobi sums and cyclotomic numbers. We then derive the cross-correlation distribution of constant multiple sequences of an M -ary power residue sequence. In the case of constant multiple sequences of an M-ary Sidel'nikov sequence, we get the possible cross-correlation values whose occurrence numbers are expressed in terms of the cyclotomic numbers of order M and are possibly zero.

New $M$-Ary Sequence Families With Low Correlation and Large Size

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> In this paper, we construct four <emphasis emphasistype="italic"><formula formulatype="inline"><tex Notation="TeX">$M$</tex></formula></emphasis>-ary sequence families from a <emphasis emphasistype="boldital">power residue sequence</emphasis> of odd prime period <emphasis emphasistype="italic"><formula formulatype="inline"> <tex Notation="TeX">$p$</tex></formula></emphasis> and its constant multiple sequences using the shift-and-add method, when <emphasis emphasistype="italic"><formula formulatype="inline"><tex Notation="TeX">$M$</tex></formula></emphasis> is a divisor of <emphasis emphasistype="italic"><formula formulatype="inline"> <tex Notation="TeX">$p-1$</tex></formula></emphasis>. We show that the maximum correlation values of the proposed sequence families are upper-bounded by <emphasis emphasistype="italic"><formula formulatype="inline"><tex Notation="TeX">$2\sqrt {p} +5$</tex></formula></emphasis> or <emphasis emphasistype="italic"><formula formulatype="inline"><tex Notation="TeX">$3\sqrt {p} +4$</tex></formula></emphasis>. In addition, we prove that the linear complexity of each sequence in the proposed families is either <emphasis emphasistype="italic"><formula formulatype="inline"> <tex Notation="TeX">$p\!-\!1$</tex></formula></emphasis> or <emphasis emphasistype="italic"><formula formulatype="inline"><tex Notation="TeX">$p\!-\!{p-1 \over M}\!-\!1$</tex> </formula></emphasis>. We also construct an <emphasis emphasistype="italic"><formula formulatype="inline"><tex Notation="TeX">$M$</tex></formula></emphasis>-ary sequence family from <emphasis emphasistype="boldital">Sidel'nikov sequences</emphasis> of period <emphasis emphasistype="italic"><formula formulatype="inline"><tex Notation="TeX">$p^m-1$</tex></formula></emphasis> by applying the same method, when <emphasis emphasistype="italic"><formula formulatype="inline"><tex Notation="TeX">$M$</tex> </formula></emphasis> is a divisor of <emphasis emphasistype="italic"><formula formulatype="inline"><tex Notation="TeX">$p^m-1$</tex></formula></emphasis>. The proposed sequence family <emphasis emphasistype="italic"><formula formulatype="inline"> <tex Notation="TeX">$\mathtilde {\cal F}_{\mmb s}$</tex></formula></emphasis> has larger size than the known <emphasis emphasistype="italic"><formula formulatype="inline"> <tex Notation="TeX">$M$</tex></formula></emphasis>-ary Sidel'nikov sequence families, whereas they all have the same upper bound on the maximum correlation. </para>

New Families of $M$-Ary Sequences With Low Correlation Constructed From Sidel'nikov Sequences

In this correspondence, for a positive integer <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</i> and a prime p such that M|p <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> - 1, three families of M-ary sequences using the <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</i> -ary Sidel'nikov sequences with period p <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> - 1 are constructed. Two small families contain [(p <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> - 1)/2]+ <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</i> -2 or p <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> + <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</i> -3 <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</i> -ary sequences, and both of their maximum magnitudes of correlation values are upper bounded by 2 radic{p <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> } + 6. The largest family has its maximum magnitude of correlation values upper bounded by 3 radic{p <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> }+5 and the family size is ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</i> -1) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> (2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n-1</sup> -1) + <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</i> -1 for p = 2 or ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</i> -1) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> (p <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> - 3)/2 + <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</i> ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</i> -1)/2 for an odd prime p.

Random $n$-ary sequence and mapping uniformly distributed

Višek [3] and Culpin [1] investigated infinite binary sequence $X=(X_1,X_2,\dots )$ with $X_i$ taking values $0$ or $1$ at random. They investigated also real mappings $H(X)$ which have the uniform distribution on $[0;1]$ (notation $\mathcal U(0;1)$). The problem for $n$-ary sequences is dealt with in this paper.

Perfect q -ary sequences from multiplicative characters over GF ( p )

A new coding rule based on multiplicative characteristics over GF (p) is identified for the construction of perfect q-ary sequences. A subset of these sequences achieves the maximum possible efficiency less than unity. The new coding rule is illustrated by demonstrating that the perfect ternary Ipatov sequences are incorporated in these perfect q-ary sequences.

A unique ranking of multilevel sequences and its application to source coding (Corresp.)

A new simple algorithm for unique ranking of q -ary sequences is presented, using a general triangle (a triangular array of integers) that includes the well-known Pascal triangle as a special case. Applying the algorithm gives a new multilevel source coding scheme for which the prefix is the Lee weight and the suffix is the ranking number. A detailed analysis of the asymptotic performance shows that the new scheme is weighted universal. The inverse enumeration algorithm for decoding is also presented, and the complexity of the algorithm increases linearly in proportion to the length of source sequences.

On the enumeration of certain sets of planted plane trees

Using the definition of planted plane trees given by D. A. Klarner (“A correspondence between sets of trees,” Indag. Math. 31 (1969), 292–296) the number of nonisomorphic classes of certain sets of these trees is enumerated by obtaining a one-to-one correspondence between these classes and certain sets of nondecreasing vectors with integral components. A one-to-one correspondence between sets of ( r + 1)-ary sequences and a certain set of planted plane trees is also established, which permits enumeration of this set. Finally, a natural generalization of Klarner's one-to-one correspondence between the above sets of trees and certain sets of edge-chromatic trees is obtained.

On the expected duration of a search for a fixed pattern in random data (Corresp.)

An expression is obtained for the expected duration of a search to find a given <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</tex> -ary sequence in a semi-infinite stream of random <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</tex> -ary data. The search time is found to be an increasing function of the lengths of the "bifices" of the pattern, where the term bifix denotes a sequence which is both a prefix and a suffix.

ON THE PROBLEM OF CLASSIFICATION OF POLYNOMIAL ENDOMORPHISMS OF THE PLANE

The paper is a continuation of the author's paper [5] (Math. USSR Sbornik 6 (1968), 97-114).§1 concerns the iterations of a polynomial of degree on a singular set . It is assumed that the critical points of lie either in the domains of attraction of finite attracting cycles or at infinity. The theorems of [5] (Theorem 1 concerning the topological isomorphism of the transformation and of a shift on the space of one-sided -ary sequences with a finite number of identifications; Theorem 2: ) are generalized for the case of a disconnected .In §2 the author investigates the iterations of on the entire plane . He shows (Theorem 3) that the dynamical systems and are topologically isomorphic for sufficiently small in the case of polynomials satisfying one of the hypotheses of §1 and a certain coarse condition of nonconjugacy of the iterations of distinct critical points.Hypothesis: the set of structurally stable mappings investigated in the paper is everywhere dense in the space of coefficients.Nine figures; bibliography of eight titles.