Two Classes of Codebooks Nearly Meeting the Welch Bound

Abstract

In this paper, the value of I <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">max</sub> (C) for codebooks C = C(D) constructed by certain almost difference sets D in F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x</sup> is determined and expressed in terms of Jacobi sums from which it shows that such codebooks nearly meet the Welch bound. This result is an answer of a question raised by C.Ding and T.Feng in [2]. We also present another series of codebooks which nearly meet the Welch bound, where the codebook is constructed by a subset R = F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q1</sub> ⊕ F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q2</sub> . When q <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> = q <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> , D <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ε</sub> is an almost difference set of R.