On the k-error linear complexity over $$\mathbb{F}_p$$ of Legendre and Sidelnikov sequences

Abstract

We determine exact values for the k-error linear complexity L k over the finite field F p of the Legendre sequence L of period p and the Sidelnikov sequence T of period p m − 1. The results arefor 1 ≤ k ≤ (p m − 3)/2 and L k (T ) = 0 for k ≥ (p m − 1)/2. In particular, we prove

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

“…To conclude the proof, it remains to note that by [1] we have S (q) (x) ≡ −1/2 + T (x)(x − 1) (q−1)/2 (mod (x − 1) q ), where T (1) 0.…”

“…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].…”

𝑘-error linear complexity over 𝔽 p of subsequences of Sidelnikov sequences of period (pr – 1)/3

“…Because of its construction over F q it is also natural to analyse the k-error linear complexity of (a n ) over F q which equals the linear complexity over F p in this case. We also know the following general results for t = q − 1 and t = (q − 1)/2, see [1,2] and references therein. The sequence (a n ) defined in (1.1) satisfies…”

On the k‐Error Linear Complexity of Binary Sequences Derived from the Discrete Logarithm in Finite Fields

“…The (aperiodic) autocorrelation of σ was analyzed in [15] and the linear complexity of σ was studied in [14,18]. In particular, in [1,2] the k-error linear complexity over F p of σ was investigated for r = 1. One might ask whether it can be extended to the case r ≥ 2 for the k-error linear complexity.…”