The weight distributions of two classes of binary cyclic codes

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Highlights

  • A certain system of algebraic equations over finite fields is solved.

  • A connection between the “type” and rank of the quadratic forms in certain case is applied.

  • The weight distribution of a class of binary cyclic codes is determined by using the connection.

  • We determine another class of binary cyclic codes by using the Pless power moment identities

Abstract

For two positive integers m and k, let Ce be a class of cyclic code of length 2m1 over F2 with three nonzeros γ1, γ(2k+1) and γ(2ek+1) for e=2 or 3, where γ is a primitive element of F2m. When mgcd(m,k) is odd, Kasami in 1971 determined the weight distributions of cyclic codes C2 and C3, which is the same as that of the dual of the triple error-correcting BCH code. This paper obtains the weight distributions of C2 and C3 for the case of mgcd(m,k) being even.

MSC

11T71
94B15

Keywords

Cyclic code
Weight distribution
Exponential sum
Quadratic form

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