Spherically Punctured Biorthogonal Codes

Abstract

Consider a binary Reed-Muller code RM(r,m) defined on the hypercube \BB F <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> and let all code positions be restricted to the m-tuples of a given Hamming weight b. In this paper, we specify this single-layer construction obtained from the biorthogonal codes RM(1,m) and the Hadamard codes H(m). Both punctured codes inherit some recursive properties of the original RM codes; however, they cannot be formed by the recursive Plotkin construction. We first observe that any code vector in these codes has Hamming weight defined by the weight w of its information block. More specifically, this weight depends on the absolute values of the Krawtchouk polynomials K <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> (w). We then study the properties of the Krawtchouk polynomials and show that the minimum code weight of a single-layer code RM(1,m,b) is achieved at the minimum input weight w = 1 for any . We further refine our codes by limiting the possible weights w of the input information blocks. As a result, some of the designed code sequences meet or closely approach the Griesmer bound. Finally, we consider more general punctured codes whose positions form several spherical layers.