Convolutional codes: techniques of construction
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Convolutional codes: techniques of construction Giuliano G. La Guardia
Received: 22 August 2014 / Revised: 18 October 2014 / Accepted: 2 November 2014 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2014
Abstract In this paper, we show how to construct new convolutional codes from old ones by applying the well-known techniques: puncturing, extending, expanding, direct sum, the (u|u+v) construction and the product code construction. By applying these methods, several new families of convolutional codes can be constructed. As an example of code expansion, families of convolutional codes derived from classical Bose–Chaudhuri–Hocquenghem, character codes and Melas codes are constructed. Keywords
Convolutional codes · Techniques of construction · Combining classical codes
Mathematics Subject Classification
11T71
1 Introduction Constructions of (classical) convolutional codes and their corresponding properties have been presented in the literature (Forney Jr 1970; Massey and Costello 1971; Piret and Krol 1983; Piret 1988a, b; Rosenthal and Smarandache 1998; Rosenthal and York 1999; Hole 2000; Smarandache et al. 2001; Massey and Costello 2001; Gluesing-Luerssen et al. 2006; Gluesing-Luerssen and Schmale 2006, 2003; Aly et al. 2007a; Gluesing-Luerssen and Tsang 2008; Hurley 2009; La Guardia 2012, 2013a, 2014a, b; O’Shaughnessy 2014). Forney Jr (1970) was the first author who introduced algebraic tools to describe convolutional codes. Addressing the construction of maximum-distance-separable (MDS) convolutional codes (in the sense that the codes attain the generalized Singleton bound introduced in Rosenthal and Smarandache 1998, Theorem 2.2), there exist interesting papers in the literature (Rosenthal
Communicated by Eduardo Souza de Cursi. G. G. La Guardia (B) Department of Mathematics and Statistics, State University of Ponta Grossa (UEPG), Ponta Grossa, PR 84030-900, Brazil e-mail: [email protected]
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and Smarandache 1998; Smarandache et al. 2001; Gluesing-Luerssen and Schmale 2003). Concerning the optimality with respect to other bounds, we have Piret and Krol (1983) and Piret (1988b), and in Gluesing-Luerssen et al. (2006), strongly MDS convolutional codes were constructed. In Rosenthal and York (1999), Hole (2000), Aly et al. (2007a) and La Guardia (2012), the authors presented constructions of convolutional BCH codes. In Gluesing-Luerssen and Schmale (2006), doubly cyclic convolutional codes were constructed and in Gluesing-Luerssen and Tsang (2008), the authors described cyclic convolutional codes by means of the matrix ring. It is not simple to derive families of such codes by means of algebraic approaches. In other words, most of the convolutional codes available in the literature are constructed case by case. To derive new families of convolutional codes by means of algebraic methods, this paper is devoted to construct new convolutional codes from old ones. More precisely, we show how to obtain new codes by extending, puncturing, expanding, applying the direct su
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