entropy Article Skew Convolutional Codes Vladimir Sidorenko 1,* , Wenhui Li 2, Onur Günlü 3 and Gerhard Kramer 1 1 Institute for Communications Engineering, Technical University of Munich, 80333 München, Germany;
[email protected] 2 Skolkovo Institute of Science and Technology, 143026 Moscow, Russia;
[email protected] 3 Information Theory and Applications Chair, Technical University of Berlin, 10623 Berlin, Germany;
[email protected] * Correspondence:
[email protected] Received: 30 October 2020; Accepted: 30 November 2020; Published: 2 December 2020 Abstract: A new class of convolutional codes, called skew convolutional codes, that extends the class of classical fixed convolutional codes, is proposed. Skew convolutional codes can be represented as periodic time-varying convolutional codes but have a description as compact as fixed convolutional codes. Designs of generator and parity check matrices, encoders, and code trellises for skew convolutional codes and their duals are shown. For memoryless channels, one can apply Viterbi or BCJR decoding algorithms, or a dualized BCJR algorithm, to decode skew convolutional codes. Keywords: convolutional codes; skew polynomials; time-varying codes; dual codes; trellises 1. Introduction Convolutional codes were introduced by Elias in 1955 [1]. With the discovery that convolutional codes can be decoded with Fano sequential decoding [2], Massey threshold decoding [3], and, above all, Viterbi decoding [4], they became quite widespread in practice. Convolutional codes are still widely used in telecommunications, e.g., in Turbo codes [5] and in the WiFi IEEE 802.11 standard [6], in cryptography [7], etc. The most common are binary convolutional codes; however, communication with higher orders of modulation [8] or streaming of data [9] require non-binary convolutional codes.