IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 56, NO. 12, DECEMBER 2010 6347 Constructing Coset Codes With Optimal Same-Symbol Weight for Detecting Narrowband Interference in -FSK Systems Daniel J. J. Versfeld, Member, IEEE, A. J. H. Vinck, Fellow, IEEE, James N. Ridley, and Hendrik C. Ferreira, Senior Member, IEEE Abstract—Narrowband interference can cause undetected errors when -FSK data is encoded with an algebraic code containing the all- codewords. This is due to the fact that the narrowband interferer will cause the output of the -FSK demodulator to correspond to one of the all- codewords. One possible solution is to use a coset code of a code containing the rep- etition code. The choice of the coset leader should be such that the resulting coset code has minimum same-symbol weight. We give a general construction for generating coset codes with minimum same-symbol weight and present results where an optimal coset code for an Reed-Solomon code is applied in an -FSK environment with narrowband interference. From the results it is evident that the optimal coset codes outperform linear codes when narrowband interference is present. Index Terms—Coset codes, minimum same-symbol weight, Reed-Solomon codes. I. INTRODUCTION O N many communication channels narrowband noise compromises the integrity of the data transmitted over the channel (see [1] and [2]). One such an environment is the Power Lines Communications (PLC) channel. In the presence of narrowband noise, algebraic codes which contain the all- codewords in conjunction with -FSK modulation suffers from undetectable errors. In [3], a method was described to detect and correct the presence of narrowband noise in -FSK systems by uti- lizing certain Reed-Solomon codes and coset Reed-Solomon Manuscript received August 26, 2008; revised July 23, 2010. Date of current version November 19, 2010. This work was supported by Telkom SA Ltd. and by the South African National Research Foundation under Grant 2053408. The material in this paper was presented (in part) at the Ninth International Sym- posium on Communication Theory and Applications (ISCTA’07), Ambleside, U.K., July 2007. D. J. J. Versfeld is with the School of Electrical and Information Engineering, University of the Witwatersrand, Johannesburg, South Africa (e-mail: jaco.vers- feld@wits.ac.za). A. J. Han Vinck is with the Institute for Experimental Mathematics, Univer- sity of Duisberg-Essen, 45326 Essen, Germany (e-mail: vinck@exp-math.uni- essen.de). J. N. Ridley is with the John Knopfmacher Center, School of Mathe- matics, University of the Witwatersrand, Johannesburg, South Africa (e-mail: james@ridley.za.net). H. C. Ferreira is with the Department of Electrical and Electronic Engi- neering, University of Johannesburg, Johannesburg, South Africa (e-mail: hcferreira@uj.ac.za). Communicated by I. Dumer, Associate Editor for Coding Theory. Digital Object Identifier 10.1109/TIT.2010.2079013 codes. In this paper we give a general algebraic construc- tion for coset codes that perform optimally in the presence of narrowband interference. We also apply the construction to an Reed-Solomon code and simulate and compare the performance of the constructed coset code with normal Reed-Solomon codes in the presence of narrowband noise and additive white Gaussian noise. The optimal coset codes out- perform Reed-Solomon codes when narrowband interference is present due to errors which the Reed-Solomon codes cannot detect. The remainder of this paper is structured as follows. Section II provides a brief overview on the effect of narrowband noise on -FSK modulated transmission. In Section III, we give a construction for finding the optimal coset code of an algebraic code. Section IV discusses how encoding and decoding can be achieved. In Section V, the results are presented and we con- clude in Section VI. II. -FSK AND NARROW-BAND NOISE We refer the reader to [4], [5], and [6] for the normal AWGN channel for -FSK modulation. We use the same one-to-one mapping from the field onto the distinct frequencies of the -FSK modulator as is done in [4] and [5]. In essence, noncoherent -FSK detection chooses from a set of frequencies the one with the highest energy present at a sampling instance , assuming that the desired frequency was transmitted with energy . Furthermore, the SNR for such a system is given as (refer to [6]). Practically noncoherent -FSK detection is implemented by using a bank of correlators, with a quadrature pair for each frequency. For each quadrature pair the output is added together using the square law to produce a metric for the corresponding frequency candidate. The most likely transmitted symbol for sampling in- stance is determined based on these metrics. For normal envelope detection, the symbol corresponding to the metric with the highest value is chosen as the candidate [6, p. 258]. The Viterbi threshold ratio test detector [7] performs a rudi- mentary form of soft-decision detection. As such, for sampling instance , the output is given as (1) where is the symbol corresponding to the trans- mitted frequency . For sampling instance is the largest metric (corresponding to frequency ) and is the second 0018-9448/$26.00 © 2010 IEEE