Perfect Gaussian Integer Sequences of Odd Period <formula formulatype="inline"><tex Notation="TeX">${2^m} - 1$</tex> </formula>

Abstract

In this letter, some perfect Gaussian integer sequences of period 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> - 1 are proposed based on the trace representations of Legendre sequences, Hall's sextic residue sequences, m-sequences, and Gordon-Mills-Welch (GMW) sequences over the finite field \BBF <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> . Moreover, the energy efficiency of these sequences is approximately 1 for sufficiently large m.