Applications Of Exponential Sums In Cycliccodes And Other Fields

Exponential sum is an important branch in number theory. In this paper, we mainly study the applications of exponential sums in cyclic codes and other fields, which contain weight distributions of cyclic codes, Walsh transforms of monomial functions, codebooks, signal sets, and strongly regular graphs.The weight distributions of cyclic codes are very important in coding and decoding theory. In this thesis, we mainly investigate the weight distributions of cyclic codes whose duals have two zeros of distinct orders, use Gauss periods and Gauss sums, first present the weight distributions of these cyclic codes, and break through the conventional case of the same order. Let qF be a finite field with q elements, mr =q, 1 2,rg g ∈ F, 1 1o(rd g) =n, 2 2o(rd g) =n, 1 2n ≠n, 1 2d =gcd(n, n),1 2n nnd=, and /Trr q the trace function from rF to qF. We define a cyclic code(,) :, }r=Cc a b a b ∈ F, where0 0 1 1 1 1/ 1 2 1 2 /1 2/(,)(Tr(), Tr(),, Tr())n n r rq q r qc a b ag bg ag bg ag bg--++= + …. In this paper, we use Gauss periods to study the weight distributions of the following cyclic codes:(1) 1 1()ord g =n=n, n| r-1, and 2g =1;(2) 1 1()ord g =n=n, 22 1g =g, 2 2()ord2ng =n=, m =2, and 2( 1)|( 1)rqn-+;(3) 1 1o(rd g) =n, 2 2o(rd g) =n, and 1 2gcd(n, n) =1;(4) 1 1o(rd g) =n, 2 2o(rd g) =n, and 1 2gcd(n, n) =d, in particular, we explicitly present the weight distributions of cyclic codes when 111mn=q-, 221mn=q-, 1 2gcd(m, m) =1.Additionally, we also use Gauss sums to determine the weight distributions of cyclic codes in the following case :111mn=q-, 221mn=q-, 1 2gcd(m, m) =2, and they are four-weight, in particular, we obtain a class of three-weight cyclic codes. Furthermore, for a class of cyclic codes from lF conjugates, we shall give their new trace representations and use exponential sums and cyclotomic numbers to determine their weight distributions.The Walsh transform is an important tool to investigate the properties of cryptographic functions. An interesting problem is to find functions that have only a few Walsh transform values and determine their distributions. The value distributions of the Walsh transforms of a class of monomials1/() Tr()rNr pf x ax-= shall be presented under some conditions and show that their Walsh transforms take a few distinct values. In particular, we can obtain a class of binary functions with three-valued Walsh transform and a class of ternary functions with three-valued or four-valued Walsh transform. Furthermore, we present two classes of four-weight binary cyclic codes and six-weight ternary cyclic codes.Codebooks and signal sets are widely applied in code-division multiple-access systems and radar and sonar systems. We shall give a new class of almost difference sets by using partial difference sets and use them to construct nearly optimal codebooks. We also present a general construction of codebooks from partial difference sets and obtain several classes of nearly optimal codebooks. For signal sets, the explicit maximum cross ambiguity amplitudes of five classes of unit time-phase signal sets will be determined by Gauss sums.Strongly regular graphs play an important role in graph theory and are equivalent to partial difference sets under some conditions. We shall construct some families of strongly regular graphs on finite fields by using unions of cyclotomic classes and index 2 Gauss sums. Furthermore, eight new infinite families of strongly regular graphs are found.