Xiang-dong Hou

Cubic bent functions

Abstract For any Boolean function f on GF(2) m , we define a sequence of ranks r i ( f ), 1 ⩽ i ⩽ m , which are invariant under the action of the general linear group GL( m , 2). If f is a cubic bent function in 2 k variables, we show that when r 3 ( f )⩽ k , f is either obtained from a cubic bent function in 2 k − 2 variables, or is in a well-known family of bent functions. We also determine all cubic bent functions in eight variables.

Permutation polynomials over finite fields - A survey of recent advances

Permutation polynomials over finite fields constitute an active research area in which advances are being made constantly. We survey the contributions made to this area in recent years. Emphasis is placed on significant results and novel methods.

Results on Bent Functions

In this paper, we present three results on bent functions: a construction, a restriction, and a characterization. Starting with a single bent function, in a simple but very effective way, the construction produces a large number of new bent functions in the same number of variables. The restriction imposes new conditions on the directional derivatives of bent functions. Certain non-existence results that were previously obtained through computer search follow easily from these conditions. The characterization describes bent functions as certain solutions of a system of quadratic equations. Interesting new properties of bent functions are obtained using the characterization.

GL ( m ,2) acting on R ( r,m )/ R ( r −1, m )

Abstract Let R ( r , m ) be the r th order Reed-Muller code of length 2 m , and let R ( r , m )/ R ( r − 1, m ) be the set of all cosets of R ( r − 1, m ) in R ( r , m ). The general linear group GL ( m ,2) acts on R ( r , m )/ R ( r − 1, m ). We compute the numbers of the GL ( m ,2)-orbits of R ( r , m )/ R ( r − 1, m ) for 6 ⩽ m ⩽ 11. This is done through a formula for the size of the centralizer of a matrix in GL ( m ,2) and the observation that any A ∈ GL ( m ,2) acts on R ( r , m )/ R ( r − 1, m ) as a linear isomorphism whose matrix with respect to the basis { Π i ∈ S X i + R ( r − 1, m ) : S ⊂ {1, …, m }, | S | = r } of R ( r , m )/ R ( r − 1, m ) is the r th compound of A . We then classify R (3, 6)/ R (2, 6) and R (3, 7)/ R (2, 7). The implication of these classifications concerning the covering properties of R (2,6) and R (2,7) is also given.

Determination of a type of permutation trinomials over finite fields, II

Let q be a prime power. We determine all permutation trinomials of F q 2 of the form a x + b x q + x 2 q - 1 ? F q 2 x . The subclass of such permutation trinomials of F q 2 with a , b ? F q was determined in a recent paper 6.

On the norm and covering radius of the first-order Reed-Muller codes

Let /spl rho/(1,m) and N(1,m) be the covering radius and norm of the first-order Reed-Muller code R(1,m), respectively. It is known that /spl rho/(1,2k+1)/spl les/lower bound [2/sup 2k/-2/sup (2k-1/2)/] and N(1,2k+1)/spl les/2 lower bound [2/sup 2k/-2/sup (2k-1/2)/] (k>0). We prove that /spl rho/(1,2k+1)/spl les/2 lower bound [2/sup 2k-1/-2/sup (2k-3/2)/] and N(1,2k+1)/spl les/4 lower bound [2/sup 2k-1/-2/sup (2k-3/2)/] (k>0). We also discuss the connections of the two new bounds with other coding theoretic problems.

A new approach to permutation polynomials over finite fields

Let p be a prime and q=pκ. We study the permutation properties of the polynomial gn,q∈Fp[x] defined by the functional equation ∑a∈Fq(x+a)n=gn,q(xq−x). The polynomial gn,q is a q-ary version of the reversed Dickson polynomial in characteristic 2. We are interested in the parameters (n,e;q) for which gn,q is a permutation polynomial (PP) of Fqe. We find several families of such parameters and obtain various necessary conditions on such parameters. Initial results, both theoretical and numerical, indicate that the class gn,q contains an abundance of PPs over finite fields, many of which are yet to be explained and understood.

Bent Functions, Partial Difference Sets, and Quasi-FrobeniusLocal Rings

Bent functions andpartial difference sets have been constructed from finite principalideal local rings. In this paper, the constructions are generalizedto finite quasi-Frobenius local rings. Let R bea finite quasi-Frobenius local ring with maximal ideal M.Bent functions and certain partial difference sets on M } M are extended to R } R.

A piecewise construction of permutation polynomials over finite fields

Abstract We describe a piecewise construction of permutation polynomials over a finite field F q which uses a subgroup of F q ⁎ , a “selection” function, and several “case” functions. Permutation polynomials obtained by this construction unify and generalize several recently discovered families of permutation polynomials.

The Reed-Muller code R(r,m) is not Z/sub 4/-linear for 3/spl les/r/spl les/m-2

After a discussion of automorphisms of Reed-Muller codes the authors show that the Reed-Muller code R(r,m) is not Z/sub 4/-linear for 3/spl les/r/spl les/m-2, proving a conjecture by Hammons, Kumar, Calderbank, Sloane, and Sole (1994).