New code equivalence based on relative generalized Hamming weights

Authors:
Zihui Liu;Xin-Wen Wu;Yuan Luo;Wende Chen
Affiliations:
Department of Mathematics, Beijing Institute of Technology, Beijing 100081, China;School of Information and Communication Technology, Griffith University, Gold Coast, QLD 4222, Australia;Department of Computer Sciences and Engineering, Shanghai Jiao Tong University, Shanghai 200240, China;Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100080, China
Venue:
Information Sciences: an International Journal
Year:
2011

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Abstract

Code equivalence is a basic concept in coding theory. The well-known theorem by MacWilliams gives a sufficient condition for code equivalence. Recently the MacWilliams theorem has been generalized, by Fan, Liu and Puig, making use of the generalized Hamming weights (GHWs). In this paper, we will present a further generalization of the MacWilliams theorem. Our result extends both the MacWilliams theorem and the result by Fan, Liu and Puig. We will first define ''relative subcodes'' of a linear code, based on the relative generalized Hamming weights (RGHWs) which is a generalization of the GHWs; and then establish a method based on finite projective geometry to characterize relative subcodes. Using this method, we will prove our main result.