On the k-error linear complexity of cyclotomic sequences

Abstract

Abstract. Exact values and bounds on the k-error linear complexity of p-periodic sequences which are constant on the cyclotomic classes are determined. This family of sequences includes sequences of discrete logarithms, Legendre sequences and Hall's sextic residue sequence.

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

“…So, by (6) S t (α v ) = 0 if and only if v ∈ H (q) 0 and q ≡ 1 (mod p) and q ≡ 1 (mod 4) or q ≡ −1 (mod p) and q ≡ 3 (mod 4). If S t (α v ) = 0 then by Lemma 4 and Corollary 6 we obtain that α v is a root of S t (x) of multiplicity p. The conclusion of this theorem then follows from (2).…”

“…Then, according to (2), (3), in order to find the minimal polynomial and the linear complexity of {s i } over GF(q) or GF(p) it is sufficient to find the roots of S (x) in the set {1, α, . .…”

“…Also, there are a long lot of articles devoted to the investigation the linear complexity of of Ding-Helleseth sequences of another orders over the different finite fields when a field's order and a sequence's period are relatively prime. At the same time, the linear complexity of cyclotomic sequences with period p over GF(p) was investigated in [1,2].…”

A Note on Hall’s Sextic Residue Sequence: Correlation Measure of Order $k$ and Related Measures of Pseudorandomness

“…Theorem 1 Let H = {h 0 , h 1 , · · · , h p−1 } be a period of Hall's sextic sequence defined by (2). Then the correlation measure of order k of H satisfies…”

On the linear complexity of Hall’s sextic residue sequences over $${ GF}(q)$$ G F ( q )

“…The linear complexity of generalized cyclotomic binary sequences of order 2 was investigated in [4] and linear complexity of Ding-Helleseth sequences of order 2 over GF(l) in [17]. The linear complexity of Hall's sextic residue sequences with the length p was considered in [14] over the finite field of order 2 and in [1] over the finite field of p elements. The linear complexity of other binary cyclotomic sequences of order 6 over the finite field of order 2 was investigated in [3,7,15] (also, see references therein).…”