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Designs, Codes and Cryptography

, Volume 74, Issue 3, pp 533–557 | Cite as

On the bounds and achievability about the ODPC of \(\mathcal{GRM }(2,m)^*\) over prime fields for increasing message length

  • Xiaogang Liu
  • Yuan LuoEmail author
Article

Abstract

The optimum distance profiles of linear block codes were studied for increasing or decreasing message length while keeping the minimum distances as large as possible, especially for Golay codes and the second-order Reed–Muller codes, etc. Cyclic codes have more efficient encoding and decoding algorithms. In this paper, we investigate the optimum distance profiles with respect to the cyclic subcode chains (ODPCs) of the punctured generalized second-order Reed–Muller codes \(\mathcal{GRM }(2,m)^*\) which were applied in Power Control in OFDM Modulations, in channels with synchronization, and so on. For this, two standards are considered in the inverse dictionary order, i.e., for increasing message length. Four lower bounds and upper bounds on ODPC are presented, where the lower bounds almost achieve the corresponding upper bounds in some sense. The discussions are over nonbinary prime fields.

Keywords

Cyclic code Exponential sum Generalized Reed–Muller code  Optimum distance profile Quadratic form 

Mathematics Subject Classification

94B05 94B65 

Notes

Acknowledgments

This work was supported in part by the National Basic Research Program of China under Grants 2012CB316100, 2013CB338004, and the National Natural Science Foundation of China under Grant 61271222. We would also like to acknowledge the foundation TS0520103001 of Shanghai Jiao Tong University and University of Leuven.

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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of Computer Science and EngineeringShanghai Jiao Tong UniversityShanghaiChina
  2. 2.National Mobile Communications Research LaboratorySoutheast UniversityNanjingChina

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