A method for improving the code rate and error correction capability of a cyclic code
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A method for improving the code rate and error correction capability of a cyclic code Tariq Shah · Amanullah · Antonio Aparecido de Andrade
Received: 30 January 2012 / Revised: 9 September 2012 / Accepted: 9 September 2012 / Published online: 16 April 2013 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2013
Abstract Currently, there has been an increasing demand for operational and trustworthy digital data transmission and storage systems. This demand has been augmented by the appearance of large-scale, high-speed data networks for the exchange, processing and storage of digital information in the different spheres. In this paper, we explore a way to achieve this goal. For given positive integers n, r , we establish that corresponding to a binary cyclic code k k C0 [n, n − r ], there is a binary cyclic code C[(n + 1)3 − 1, (n + 1)3 − 1 − 3k r ], where k is a nonnegative integer, which plays a role in enhancing code rate and error correction capability. In the given scheme, the new code C is in fact responsible to carry data transmitted by C0 . Consequently, a codeword of the code C0 can be encoded by the generator matrix of C and therefore this arrangement for transferring data offers a safe and swift mode. Keywords
Polynomial ring · Semigroup ring · Binary cyclic code · Binary Hamming code
Mathematics Subject Classification
18B40 · 94A15 · 94B05
1 Introduction Principal ideals in a unitary commutative ring are very indispensable in applications of finite algebraic structures. The coding for error control has essential fragment in the scheme of
Communicated by José Eduardo Souza de Cursi. T. Shah · Amanullah Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan e-mail: [email protected] Amanullah e-mail: [email protected] A. A. de Andrade (B) Department of Mathematics, São Paulo State University, São José do Rio Preto, SP, Brazil e-mail: [email protected]
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current communication systems and a high rate data exchange in digital computers. Most of the conventional error-correcting codes are ideals in finite principal ideal rings, particularly in factor rings F[X ]n , where F is a field. In Cazaran and Kelarev (1997), the authors described all finite commutative principal ideal rings mZZ [Y1 , . . . , Yt ]/L, where m and t are positive integers and L is an ideal generated by univariate polynomials. Nevertheless, in Cazaran and Kelarev (1999), the authors found conditions for certain unitary commutative rings to be finite principal ideal rings. Moreover, in Cazaran et al. (2006), the authors considered the extension of a BCH code embedded in a semigroup ring. A survey of information related to various ring constructions used to investigate codes is given in Kelarev (2002), see also Kelarev and Solé (2001). In Andrade and Palazzo (2005), the authors constructed cyclic, BCH, alternant, Goppa and Srivastava codes via the parity-check matrix, where the entries are units of a Galois ring extension of a finite local ring; whereas, the
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