Sequences and Linear Codes from Highly Nonlinear Functions
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Due to optimal nonlinearity and differential uniformity, perfect nonlinear (PN) and almost perfect nonlinear (APN) functions are of great importance in cryptography. It is interesting that they also define optimal objects in other domains of mathematics and information theory. This dissertation is devoted to exploring the application of highly nonlinear functions, especially PN and APN functions, to the construction of low-correlation sequences and optimal linear codes. For an arbitrary odd prime p, there are only two basic classes of two-level auto-correlation p-ary sequences with no subfield structures: the msequences and the Helleseth-Gong sequences, where Helleseth-Gong sequences are closely related to a class of p-ary perfect nonlinear functions. Papers I and II are dedicated to investigating the cross-correlation between the p-ary m-sequences and d-decimated Helleseth-Gong sequences for some decimations d, and to constructing sequence families with low correlation from them. Papers III-IV have focused on the study of linear codes defined from highly nonlinear functions. Paper III utilizes some highly nonlinear functions including PN and APN functions to construct ternary cyclic codes with the optimal minimum (Hamming) distance. Paper IV further investigates the weight distribution of some optimal cyclic codes proposed in Paper III. Paper V examines the covering radius of some linear codes defined from PN and APN functions and presents a number of quasi-perfect linear codes.
Består avPaper I: Guang Gong, Tor Helleseth, Honggang Hu and Chunlei Li,“New Three-Valued Walsh Transforms from Decimations of Helleseth- Gong Sequences", Proceedings of SETA 2012: The 7th International Conference on SEquences and Their Applications. LNCS 7280, pp. 327-337, 2012. Full text not available in BORA due to publisher restrictions. The article is available at: http://dx.doi.org/10.1007/978-3-642-30615-0_30.
Paper II: Chunlei Li and Tor Helleseth, “New Nonbinary Sequence Families with Low Correlation and Large Linear Span", Proceedings of ISIT 2012: International Symposium on Information Theory. IEEE Press 2012, pp. 1411-1415, 2012. Full text not available in BORA due to publisher restrictions. The article is available at: http://dx.doi.org/10.1109/ISIT.2012.6283494.
Paper III: Nian Li, Chunlei Li, Tor Helleseth, Cunsheng Ding and Xiaohu Tang, “Optimal Ternary Cyclic Codes with Minimum Distance Four and Five", Finite Fields and Their Applications, vol. 30: 100-120, 2014. The article is available at: http://hdl.handle.net/1956/8429.
Paper IV: Chunlei Li, Nian Li, Tor Helleseth and Cunsheng Ding, “The Weight Distributions of Several Classes of Cyclic Codes from APN Monomials", accepted by IEEE Transaction on Information Theory, 60(8): 4710-4721, 2014. The article is available at: http://hdl.handle.net/1956/8430.
Paper V: Chunlei Li and Tor Helleseth, “Quasi-Perfect Linear Codes from Planar and APN functions". Full text not available in BORA.