Design of Length-Compatible Polar Codes Based on the Reduction of Polarizing Matrices

Abstract

Length-compatible polar codes are a class of polar codes which can support a wide range of lengths with a single pair of encoder and decoder. In this paper we propose a method to construct length-compatible polar codes by employing the reduction of the 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> × 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> polarizing matrix proposed by Arikan. The conditions under which a reduced matrix becomes a polarizing matrix supporting a polar code of a given length are first analyzed. Based on these conditions, length-compatible polar codes are constructed in a suboptimal way by codeword-puncturing and information-refreezing processes. They have low encoding and decoding complexity since they can be encoded and decoded in a similar way as a polar code of length 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> . Numerical results show that length-compatible polar codes designed by the proposed method provide a performance gain of about 1.0 - 5.0 dB over those obtained by random puncturing when successive cancellation decoding is employed.