The Cardinality of the Set of Zeros of Homogeneous Linear Recurring Sequences over Finite Fields

Consider homogeneous linear recurring sequences over a finite field Fq, based on the irreducible characteristic polynomial of degree d and order m. We give upper and lower bounds, and in some cases the exact values of the cardinality of the set of zeros of the sequences within its least period. We also prove that the cyclotomy bound introduced here is the best upper bound as it is reached in infinitely many cases. In addition, the exact number of occurrences of zeros is determined using the correlation with irreducible cyclic codes when (qd − 1)/m follows the quadratic residue conditions and also when it has the form q2a − qa + 1 where a ∈ N.


Introduction
Let F q be the finite field with q elements where q = p m for prime p.Let k be a positive integer, and let a 0 , a 1 , . . ., a k−1 be given elements of F q .A sequence s 0 , s 1 , . . . of elements of F q satisfying the relation s n+k = a k−1 s n+k−1 + a k−2 s n+k−2 + . . .+ a 0 s n f or n = 0, 1, . . .
is called a (kth-order) homogeneous linear recurring sequence in F q .The terms s 0 , s 1 , . . .s k−1 , which determine the complete sequence uniquely, are referred to as the initial values.A relation in the form of (1) is called a (kth-order) homogeneous linear recurrence relation.Let s 0 , s 1 , . . .be a kth order homogeneous linear recurring sequence in F q satisfying the linear recurrence relation (1), where a j ∈ F q for 0 ≤ j ≤ k − 1.The polynomial is called a characteristic polynomial of the linear recurring sequence.For s 0 , s 1 , . . .homogeneous linear recurring sequence in F q , m(x) ∈ F q [x] is said to be the minimal polynomial of the sequence if it has the following property: a monic polynomial f (x) ∈ F q [x] of positive degree is a characteristic polynomial of s 0 , s 1 , . . .if and only if m(x) divides f (x).
Definition Let f ∈ F q [x] be a non zero polynomial.If f (0) 0, then the least positive integer e for which f (x) divides x e − 1 is called the order of f which is denoted by ord( f ).
Theorem 1. (Lidl & Niederreiter, 1994) Let s 0 , s 1 , ... be a homogeneous linear recurring sequence in F q with minimal polynomial m(x) ∈ F q [x].Then the least period of the sequence is equal to ord(m(x)).
Linear recurring sequences were discussed for many years with a substantial development in the study of examining zeros and determining effective bounds for the set of zeros over infinite fields (Everest, Poorten, Shparlinski & Ward, 2003).Here we will consider homogeneous linear recurring sequences over finite fields based on irreducible minimal polynomials of certain degree d and order m.Let P(d, m) be the set of all irreducible polynomials over F q of degree d and order m.For f ∈ P(d, m) and I ∈ (F d q ) * = (F d q ) \ {0}, let S (I, f ) := {s n (I, f )|1 ≤ n ≤ m} be the first m terms (terms within the least period) of the homogeneous linear recurring sequence S over F q .Let A := {Z(S (I, f ))|I ∈ (F d q ) * , f ∈ P(d, m)} be the set of zeros.Let t = (q d − 1)/m.We will always assume that t > 1.If t = 1 then the polynomials in P (d, m) are primitive and the number of zeros in the sequence is q d−1 − 1 (Lidl & Niederreiter, 1994).However, in the general case such an equitable distribution of zeros cannot be expected.Theorem 6.84 in Lidl and Niederreiter (1994) provides an estimate for the number of occurrences of zeros based on Gaussian sums and Mullen and Panario (2013) provides an improved bound.Table 1 gives some observations on the number of zeros of some linear recurring sequences over F 2 computed via MAPLE (Kottegoda, 2010, Appendix I-VIII) with the degrees and orders of their corresponding irreducible minimal polynomials.In this paper, in addition to explaining why there are so few choices for the number of zeros, we will give an accurate bound for the cardinality of the set of zeros, also providing formulas for the exact number of zeros when t has the form q 2a − q a + 1 where a ∈ N.  411, 395, 387, 379, 363, 355 6 16 1285 669, 653, 645, 637, 621, 613, 581 7 16 4369 2225, 2185, 2177, 2169, 2097 5 Section 2 proves that the cardinality of the set of zeros is at most the number of q-cyclotomy classes in Z t , namely, the cyclotomy bound.
In section 3, results on irreducible cyclic codes are used to show | A |= 2 if t has the form q 2a − q a + 1 and also gives the exact values for A in this case.We also get a lower bound on | A | when q = 2 using results from Wolfmann (2005).Exact values for | A | when t follows the quadratic residue conditions are also discussed.Lastly, we show that the cyclotomy bound given in section 2 is the best bound as it is reached infinitely often, assuming the Generalized Riemann Hypothesis.

Construction of the Cyclotomy Bound
First we will define the following equivalence relation on Definition Let t be relatively prime to q.The cyclotomy class of q (or q-cylcotomy coset) modulo t containing i is defined by which is the equivalence class that contains i in the above mentioned equivalence relation.
Let C denote the set of all equivalence classes.The following theorem explains that when the characteristic polynomial is irreducible, a suitable trace form can be used to represent the terms of the linear recurring sequence S .
Theorem 2. (Lidl & Niederreiter, 1994) Let s 0 , s 1 , . . .be a kth-order homogeneous linear recurring sequence in K = F q whose characteristic polynomial f (x) is irreducible over K. Let α be a root of f (x) in the extension field F = F q k .Then there exists a uniquely determined θ ∈ F such that Theorem 3 below gives the upper bound for the cardinality of the set of zeros.
Theorem 3. Consider the homogeneous linear recurring sequences over F q based on an irreducible minimal polynomial of degree d and order m.Set t = (q d − 1)/m.Then for the set of numbers of zeros A, we have Proof.
Let f ∈ P (d, m).By Theorem 2, there exists a root of f , β ∈ F q d and θ ∈ F * q d such that the nth term of the sequence S is given by, Fix a primitive element α ∈ F q d .Then order of β = m and hence β = α rt where t = (q d − 1)/m and (r, m) = 1.Define Therefore, Proof.Since (r, m) = 1, there exists a u such that ur ≡ 1 (mod m) and then urt ≡ 1 (mod q n − 1).Hence Now A in (2) can be given as follows: Hence A in (3) can be written as the following where R denotes the sequence r 1 , r 2 , . . .
Using Lemma 2, A in (4) can be given as follows:

Properties of Cyclotomy Classes
Here we discuss some properties of the cyclotomy classes where we will be able to find the exact value for the cyclotomy bound and give the exact upper bound for the cardinality of the set of zeros | A |, under specific conditions.Let ord a (b) be the smallest positive integer c such that a c ≡ 1 (mod a).By the equivalence relation defined in section 2, C 1 = {1, q, q 2 , . . ., q k−1 } (mod t) where k = ord t (q).Hence | C 1 |= ord t (q).Proposition 1.If t is a composite and l | t, then there exists C l ∈ C.
Proof.Let l ∈ C a for some a ∈ Z t .Then by the definition of C a , l ≥ a and l ≡ q r a (mod t) for some r ∈ Z. Since Proposition 2. Let t ∈ N and t and q are relatively prime.Then Let t be a prime power (say p k ) where p is an odd prime.If 2 is a primitive root of Z * p 2 , then | C |= k + 1.

Coding Theory Approach
Weight distributions of irreducible cyclic codes were studied by Baumert and McEliece (1972), Baumert and Mykkeltveit (1973), Aubrey and Langevin (2005), Wolfmann (2005), Vega (2007), Aubrey and Langevin (2008) and Ding (2009).We will use these results to determine the exact occurrences of zeros in some cases, and determine the cardinality of the set of zeros of homogeneous linear recurring sequences based on irreducible minimal polynomials of fixed degree and order.First we set notations and review the basic facts as found on Lidl and Niederreiter (1994).
Let f (x) ∈ F q [x] be an irreducible polynomial of degree d and order m.Let S = {s n } be a homogeneous linear recurring sequence over F q based on f as its minimal polynomial.By Theorem 2, s n = T r K/F (θα n ), where where the entries represent the terms of the sequence S within its least period m.Set C(α) is then a cyclic code whose words represent the terms of each sequence S within its least period, based on f (x).Thus C(α) has length m and dimension d.The generator polynomial of C(α) is the reciprocal of (x m − 1)/ f (x), so that C(α) is in fact an irreducible cyclic code.
Note that the weight wt(c(θ, α)), the number of non zero entries of the code word c(θ, α) is m − Z(S ).The reductions of Theorem 3 show that all sequences based on irreducible minimal polynomials of degree d and order m have the same number of zeros.Hence We say a code is a N-weight code if it has N non-zero weights and hence for this case, N =| A |.

Lower Bounds for the Cardinality of the Set of Number of Zeros of Hhomogeneous Linear Recurring Sequences
Theorem 4. (Wolfmann, 2005) Let C be an [n, k] linear code over F q .If C is a 1-weight code with weight w and if the weight of the dual code is at least 2, then there exists Corollary 3. Let C be an irreducible cyclic 1-weight code with length m and dimension d.Set t = (q d − 1)/m.Then t divides q − 1.
Proof.We first check that the dual code C ⊥ does not have minimal weight one.Suppose it has a minimal weight of one.
As C ⊥ is also cyclic, the existence of a codeword of weight 1 in C ⊥ implies that all vectors of weight 1 are in C ⊥ and hence C ⊥ = F m q .But then C = {0}, which is not a 1-weight code.We can thus apply Theorem 4 to get m = λ(q d − 1)/(q − 1) for some λ.Hence q − 1 = λ(q d − 1)/m = λt.
Proof.Let f (x) be an irreducible polynomial of degree d and order m.
, where α is a root of f (x), is a 1-weight irreducible cyclic code.By Corollary 3, t divides q − 1 = 1 so that t = 1 and f is primitive.

Kasami-Welch approach
Theorem 5. (Wolfmann, 2005) Let C be an irreducible cyclic code of length m over F q .Let F q d be the splitting field of x m − 1 over F q .Let t be the integer such that mt = q d − 1.If d = 2e and if there exists a divisor r of e such that q r ≡ −1 (mod t), then C is a 2-weight code with weights w 1 = (q − 1)q e−1 ( q e + (t − 1)ϵ t ) w 2 = (q − 1)q e−1 ( q e − ϵ t ) , where ϵ is 1 or −1.
Theorem 6.Let q = 2. Consider sequences based on an irreducible, non-primitive polynomial of degree d and order m.Set t = (2 d − 1)/m.Suppose t is prime and 2 is a primitive root modulo t.Then where ϵ is 1 or -1, determined by 2 e ≡ ϵ (mod t).
Proof.C 1 is the subgroup of Z * t generated by 2, hence C 1 = Z * t .So there are exactly two cyclotomy classes, represented by 0 and 1.We have ord t (2) = t − 1 is even and 2 d ≡ 1 (mod t) so that t − 1 divides d.Write d = 2e.For r = t−1 2 we have r | e and 2 r ≡ −1 (mod t).So by Theorem 5, | A |= 2 and its values are as given.
Assuming the Generalized Riemann Hypothesis (GRH), Hooley (1967) proved the Artin Conjecture and in particular, that there are infinitely many primes t such that 2 is a primitive root modulo t.Together with Theorem 6, we thus get the following corollary that proves the cyclotomy bound determined in section 2 is the best bound for | A |.
Corollary 5. Assume the GRH.For q = 2, the cyclotomy bound is achieved infinitely often.
Suppose k has the form 2x. Then q 2a ≡ 1 (mod t) and since t = q 2a − q a + 1, we have q 2a ≡ q a − 1 (mod t).So t divides q a − 2. If q = 2, we assume that a ≥ 2 and hence q a − 2 0. Therefore, which is impossible.
Fix an irreducible polynomial f (x) ∈ F q [x] of degree d and order m.Let α be a root of f .We wish to apply Theorem 5 to C(α).Now t | q d − 1 so that the order of q modulo t, namely 6a, divides d.So d is even; write d = 2e.Set r = 3a.Then r divides e and by (2), q r ≡ −1 (mod t).Thus C(α) is a 2-weight code and | A |= 2. We have wt[c(θ, α)] = m − Z(S (θ, α)) so Theorem 5 proves the elements of | A | are as stated.
Remark 1 When F is a finite field of even characteristic, the terms of the homogeneous linear recurring sequence take the form of the well known Kasami-Welch function T r K/F (x 2 2a −2 a +1 ).
Theorem 8. Consider sequences based on an irreducible polynomial over F q of degree d and order m.Set t = (q d − 1)/m.Suppose Therefore l | a and hence l ≤ a. Hence l = a and C l ∈ C. The following well known result and the corollaries give the exact values for the cyclotomy bound | C | and hence the exact upper bound for the cardinality of the set of zeros | A |.

Table 1 .
Zeros of some homogeneous linear recurring sequences over F 2 based on degree d and order m irreducible minimal polynomials.