On The Optimum Distance Profles Of The Punctured Reed-muller Codes And The Weight Distributions Of Relevant Cyclic Codes

Cycliccodesformanimportantsubclassoflinearblockcodes. Cycliccodeshavealong history and there are many researches about them. They have applications in con-sumer electronics, data transmission technologies, broadcast systems, and computerapplications. These codes are attractive for two reasons:encoding and syndrome computation can be implemented easily by employingshift registers with feedback connections;because they have considerable inherent algebraic structures, it is possible todevise various practical methods for decoding.Cyclic codes are widely used in communication systems, and particulary efcient forerror detection. They were frst studied by Eugene Prange in1957, since then, manyclasses of cyclic codes have been constructed over the years, including BCH codes,Reed-Solomoncodes,Euclideangeometrycodes,projectivegeometrycodes,quadraticresidue codes, and Fire codes, etc. In this paper, we study cyclic codes from two stand-points, on the one hand we consider the optimum distance profles of the puncturedReed-Muller codes and the punctured generalized Reed-Muller codes, on the otherhand we investigate the weight distributions of cyclic codes.Theoptimumdistanceprofleisanewresearchfeldinlinearcodes,HanVinckandLuo studied the optimum distance profles of the Reed-Solomon codes, Golay codes,frst-order Reed-Muller codes and some other linear codes. The research on ODP canbe applied to design the transport format combination indicator (TFCI) in the commu-nication system defned by the3rd Generation Partnership Project (3GPP). This paper studies the optimum distance profiles with respect to the cyclic subcode chains of two classes of cyclic codes, and the weight distributions of some cyclic codes.Reed-Muller codes are important linear codes, after deleting the first coordinate they are cyclic, called the punctured Reed-Muller codes. Let RM(2,m) denote the second-order Reed-Muller code, and RM(2, m)*denote the punctured second-order Reed-Muller code. In2009, Chen and Han Vinck studied the optimum distance profile of the second-order Reed-Muller code, and gave a lower bound which was proved to be tight for m≤7. This paper studies the optimum distance profiles with respect to the cyclic subcode chains of the punctured second-order Reed-Muller code. Under the condition that the second selected cyclic subcode is the punctured first-order Reed-Muller code, the optimum DPC can be obtained in the inverse dictionary order under Standard II. In fact, this profile is only a lower bound on ODPC, which is verifed to be the ODPC-Ⅱinv for even m.Reed-Muller codes are binary linear codes, when the corresponding finite fields of the Boolean functions are nonbinary, the generalized Reed-Muller codes are ob-tained. They are cyclic after deleting one coordinate, called the punctured generalized Reed-Muller codes, denoted by gRM(μ,m) and gRM(μ,m)*respectively, where μ represents the order. They have applications in power control in OFDM modulations, in channeles with synchronization, and so on. This paper studies the optimum dis-tance profiles with respect to the cyclic subcode chaines of gRM(μ, m)*. Four lower bounds and upper bounds are presented in the inverse dictionary order under two stan-dards, where the lower bounds almost achieve the upper bounds in some sense.The weight distribution of a linear code is important both in theory and applica-tions:The weight distribution gives the minimum distance of a code, and thus the error correcting capability.The weight distribution of a code allows the computation of the error probability of error detection and corrrection with respect to some error detection and error correction algorithms. The weight distributions of cyclic codes give all the weights, and analyzing them in a whole, especially the minimum distance will be a help for the study of the optimum distance profile. On the contrary, the optimum distance profiles consider the minimum distances of the codes with nested structure, which can give some insight for the upper and lower bounds on the minimum distances of the subcodes.The weight distributions of some cyclic codes with three or four nonzeros are stud-ied in this paper. For the weight distributions of cyclic codes, MacWilliams and Seery gave a method for binary case, which can only implemented on powerful computers. McEliece, Rumsey and Van Der Vlugt connected this subject with the computation of some exponential sums which are generally hard to determine. Schoof studied its rela-tion with the number of rational points of some curves. Recently, relevant authors have studied the applications to this subject of cyclcotomic cosets, elliptic curves, Gaussian periods, group characters, Gauss sums, Singer difference sets,etc. For an odd integer m, we calculate the weight distributions of two classes of cyclic codes over prime field Fp and with length pm-1(p=3), by using exponential sums, cyclotomic cosests, association schemes and quadratic forms, etc.When m is an even integer, the weight distributions of a class of cyclic codes with three nonzeros and length pm-1are studied. Comparing to the odd case, we use quadratic forms, exponential sums and the computation of higher moments of ex-ponential sums. An important fact is to demonstrate the application of the computer algebra software in the computation of higher moments. At the same time, we show the application of Mac Williams’identities to relevant studies.