Linear Recurring Sequences and Explicit Factors of x2nd−1 in 𝔽q[x]

Let 𝔽q be a finite field of odd characteristic containing q elements, and n be a positive integer. An important problem in finite field theory is to factorize xn − 1 into the product of irreducible factors over a finite field. Beyond the realm of theoretical needs, the availability of coefficients of irreducible factors over finite fields is also very important for applications. In this paper, we introduce second order linear recurring sequences in 𝔽q and reformulate the explicit factorization of [Formula: see text] over 𝔽q in such a way that the coefficients of its irreducible factors can be determined from these sequences when d is an odd divisor of q + 1.

Bounds on the discrepancy of linear recurring sequences over Galois rings

We study the discrepancy of linear recurring sequences over Galois rings. By means of an estimate of an exponential sum some nontrivial bounds on the discrepancy are derived. It is shown that these bounds are asymptotically not worse than known estimates for maximal period linear recurring sequences over prime fields.

Bounds on the frequencies of tuples on parts of the period of linear recurring sequences over Galois rings

We study the frequencies of tuples in linear recurring sequences (LRS) of vectors over Galois rings. By means of an estimate of an exponential sum some nontrivial bounds on the frequencies of elements in LRS are derived. It is shown that these bounds are in some cases sharper than known results.

Congruence properties of pk(n)

We present a unified approach to establish infinite families of congruences for [Formula: see text] for arbitrary positive integer [Formula: see text], where [Formula: see text] is given by the [Formula: see text]th power of the Euler product [Formula: see text]. For [Formula: see text], define [Formula: see text] to be the least positive integer such that [Formula: see text] and [Formula: see text] the least non-negative integer satisfying [Formula: see text]. Using the Atkin [Formula: see text]-operator, we find that the generating function of [Formula: see text] (respectively, [Formula: see text]) can be expressed as the product of an integral linear combination of modular functions on [Formula: see text] and [Formula: see text] (respectively, [Formula: see text]) for any [Formula: see text] and [Formula: see text]. By investigating the properties of the modular equations of the [Formula: see text]th order under the Atkin [Formula: see text]-operator, we obtain that these generating functions are determined by some linear recurring sequences. Utilizing the periodicity of these linear recurring sequences modulo [Formula: see text], we are led to infinite families of congruences for [Formula: see text] modulo any [Formula: see text] with [Formula: see text] and periodic relations between the values of [Formula: see text] modulo powers of [Formula: see text]. As applications, infinite families of congruences for many partition functions such as [Formula: see text]-core partition functions, the partition function and Andrews’ spt-function are easily obtained.

Construction of Uniformly Distributed Linear Recurring Sequences Modulo Powers of 2

The aim of the present paper is to provide the background to construct linear recurring sequences with uniform distribution modulo 2s. The theory is developed and an algorithm based on the achieved results is given. The constructed sequences may have arbitrary large period length depending only on the computational power of the used machines.

Quadratic form Approach for the Number of Zeros of Homogeneous Linear Recurring Sequences over Finite Fields

We consider homogeneous linear recurring sequences over a finite field $\mathbb{F}_{q}$, based on an irreducible characteristic polynomial of degree $n$ and order $m$. Let $t=(q^{n}-1)/ m$. We use quadratic forms over finite fields to give the exact number of occurrences of zeros of the sequence within its least period when $t$ has q-adic weight 2. Consequently we prove that the cardinality of the set of zeros for sequences from this category is equal to two.