Autocorrelation Values of Generalized Cyclotomic Sequences with Period p n +1

Recently Edemskiy proposed a method for computing the linear complexity of generalized cyclotomic binary sequences of period <inline-formula> <math display="inline"> <semantics> <msup> <mi>p</mi> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </semantics> </math> </inline-formula>, where <inline-formula> <math display="inline"> <semantics> <mrow> <mi>p</mi> <mo>=</mo> <mi>d</mi> <mi>R</mi> <mo>+</mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula> is an odd prime, <inline-formula> <math display="inline"> <semantics> <mrow> <mi>d</mi> <mo>,</mo> <mi>R</mi> </mrow> </semantics> </math> </inline-formula> are two non-negative integers, and <inline-formula> <math display="inline"> <semantics> <mrow> <mi>n</mi> <mo>&gt;</mo> <mn>0</mn> </mrow> </semantics> </math> </inline-formula> is a positive integer. In this paper we determine the exact values of autocorrelation of these sequences of period <inline-formula> <math display="inline"> <semantics> <msup> <mi>p</mi> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </semantics> </math> </inline-formula> <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>n</mi> <mo>&#8805;</mo> <mn>0</mn> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> with special subsets. The method is based on certain identities involving character sums. Our results on the autocorrelation values include those of Legendre sequences, prime-square sequences, and prime cube sequences.

Abstract licence: CC BY

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