Research On Weight Distributions Of Linear Codes Based On Exponential Sum And Their Applications

The weight distributions of linear codes are very important in coding and decoding theory.In this thesis,the weight distributions of several linear codes are mainly investigated by using exponential sum and cyclotomic number,and several optimal cyclic codes and some almost optimal cyclic codes are investigated.Two kinds of linear codes with the length 2m are investigated in_q?37?by using quadratic functions.The weight distributions of two linear codes are presented,whereq?28?p^m,p is an odd prime and m is even.Firstly,two defining sets₀D and₁D are presented to construct linear codes)_D?34 ₀and?_D(34₁.Secondly,weight distributions of two linear codes are computed by using Weil sum,Gauss Sum and cyclotomic number.Finally,two linear codes are verified to be employed in secret sharing schemes.Three kinds of linear codes with the length mn are investigated in_q?37?by using planar functions.The weight distributions of three linear codes are presented,whereq?28?p^m,p is an odd prime and m is a positive integer.Firstly,three defining sets₂D,₃D and₄D are presented to construct linear codes_2D?34?,_3D?34?and_4D?34?.Secondly,weight distributions of three linear codes are computed by using Weil sum,Gauss Sum.Thirdly,two special linear codes are presented and the weight distribution of_6D?34?is computed.Finally,three linear codes are verified to be employed in secret sharing schemes.Two exponential sums are defined based on quadratic functions inZ₄,which can be used to construct two kinds of quaternary codes inZ₄.The hamming weights,Lee weights and complete weights of distributions of quaternary codes are presented with the value distributions of exponential sums.Several optimal cyclic codes with are constructed with generator polymialsg?7?x?8??28?m_e?7?x?8?m_-e?7?x?8?andg?7?x?8??28??7?x-1?8?m _e?7?x?8?m_-e?7?x?8?.Firstly,an optimal binary cyclic code with parametersé_?2 ^m-1,2^m-1-2 m,5ù_?has been proved exactly as the binary Melas code.Secondly,a near optimal quaternary cyclic code with parametersé_?4 ^m-1,4^m-1-2 m,3ù_?is investigated exactly as the quaternary Melas code.Finally,four optimal cyclic codes with parametersé_?2 ^m-1,2^m-2m-2,6ù_?,é_?3 ^m-1,3^m-2-2 m,5ù_?,é_?4 ^m-1,4^m-2-2 m,4ù_?,é_?5 ^m-1,5^m-2-2 m,4ù_?are constructed respectively.