2-Adic Complexity of Two Classes of Generalized Cyclotomic Binary Sequences

<jats:p> In this paper, 2-adic complexity of two classes of generalized cyclotomic binary sequences is investigated. The sequences in the first class have period <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="eq-00001.gif"><mml:mrow><mml:mi>p</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>p</mml:mi><mml:mo>+</mml:mo><mml:mn>4</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math>, and their 2-adic complexity attains the maximum, where both p and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="eq-00002.gif"><mml:mrow><mml:mi>p</mml:mi><mml:mo>+</mml:mo><mml:mn>4</mml:mn></mml:mrow></mml:math> are primes, and gcd<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="eq-00003.gif"><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>p</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mi>p</mml:mi><mml:mo>+</mml:mo><mml:mn>3</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:math>. The sequences in the second one have period <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="eq-00004.gif"><mml:mrow><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:math>, and their 2-adic complexity also reaches the maximum if p is an odd prime with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="eq-00005.gif"><mml:mrow><mml:mi>p</mml:mi><mml:menclose notation="updiagonalstrike"><mml:mo>≡</mml:mo></mml:menclose><mml:mn>5</mml:mn><mml:mo>,</mml:mo><mml:mn>19</mml:mn></mml:mrow></mml:math> (mod 24). </jats:p>