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Denote by b(n) the number of non-equivalent linear codes in F"2^n and by G"n","2 the number of subspaces in F"2^n. M. Wild gave a proof that lim"n"->"~(n!b(n)G"n","2)=1. R. Lax pointed out that Wild's proof contains a gap which does not appear to have an easy fix. In this paper, we give a complete proof for the formula lim"n"->"~(n!b(n)G"n","2)=1.

A finite commutative chain ring is a finite commutative ring whose ideals form a chain. Let R be a finite commutative ring with maximal ideal M and characteristic pn such that R/M≅GF(pr) and pR=Me, e⩽s, where s is the nilpotency of M. When (p−1)∤e, the structure of the group of units R× of R has been determined; it only depends on the parameters p,n,r,e,s. In this paper, we give an algorithmic method which allows us to compute the structure of R× when (p−1)|e; such a structure not only depends on the parameters p,n,r,e,s, but also on the Eisenstein polynomial which defines R as an extension over the Galois ring GR(pn,r). In the case (p−1)∤e, we strengthen the known result by listing a set of linearly independent generators for R×. In the case (p−1)|e but p∤e, we determine the structure of R× explicitly.

We give several necessary conditions for the reversed Dickson polynomial D"n(1,x) to be a permutation of F"q. In particular, we give explicit evaluation of @?"a"@?"F"""qD"n(1,a)^i for i=1,2.

Let 0 0. Define S(f,n)[email protected]?"x"@?"F"""2"""^"""ne(f(x)) where m|n and e(x)=(-1)^T^r^"^F^"^"^"^2^"^"^"^^^"^"^"^n^"^/^"^F^"^"^"^2^(^x^). We establish a relation among S(f,n) for all n with the same 2-adic order. When @n"2(@a"1)[email protected]"2(@a"k), where @n"2 is the 2-adic order function, we are able to compute S(f,n) explicitly for all n with a given f. Moreover, we are able to compute S(ax^2^^^@a^+^1+cx,n) explicitly for all @a>0, [email protected]?F"2"^"m, m|n and [email protected]?F"2"^"n.

Let R be a commutative ring. A power series f ∈ R [ [ x ] ] with (eventually) periodic coefficients is rational. We show that the converse holds if and only if R is an integral extension over Z m for some positive integer m . Let F be a field. We prove the equivalence between two versions of rationality in F [ [ x 1 , … , x n ] ] . We extend Kronecker’s criterion for rationality in F [ [ x ] ] to F [ [ x 1 , … , x n ] ] . We introduce the notion of sequential code which is a natural generalization of cyclic and even constacyclic codes over a (not necessarily finite) field. A truncation of a cyclic code over F is both left and right sequential (bisequential). We prove that the converse holds if and only if F is algebraic over F p for some prime p . Finally, we show that all sequential codes are obtained by a simple and explicit construction.

Reversed Dickson polynomials over finite fields are obtained from Dickson polynomials Dn(x,a) over finite fields by reversing the roles of the indeterminate x and the parameter a. We study reversed Dickson polynomials with emphasis on their permutational properties over finite fields. We show that reversed Dickson permutation polynomials (RDPPs) are closely related to almost perfect nonlinear (APN) functions. We find several families of nontrivial RDPPs over finite fields; some of them arise from known APN functions and others are new. Among RDPPs on Fq with q

An Alexander quandle is a module M over Z [ t , t − 1 ] whose quandle operation is defined by x ⁎ y = t x + ( 1 − t ) y , x , y ∈ M . The automorphism group of a general Alexander quandle, previously unknown, is determined in the present paper.

Let F be a field of characteristic p and let A and B be two Jordan blocks with nonzero eigenvalues. The reduction formulas of C.W. Norman enable us to compute the elementary divisors of A ⊗ B . We investigate connections between the elementary divisors of A ⊗ B and binomial matrices. In particular, we show that Norman’s reduction formulas can be used to compute the p -rank of certain binomial matrices.

If R is a homomorphic image of a finite Frobenius local ring, there is a known construction that produces Latin square type partial difference sets (PDS) in R×R. By a simple construction, we show that every finite ring is a homomorphic image of a finite Frobenius ring and every finite local ring is a homomorphic image of a finite Frobenius local ring. Consequently, Latin square type PDS can be constructed in R×R for any finite local ring R, where the additive group (R,+) can be any finite abelian p-group.

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.

In a previous paper [X. Hou, K.H. Leung, Q. Xiang, A generalization of an addition theorem of Kneser, J. Number Theory 97 (2002) 1–9], the following result was established: let E⊂K be fields such that the algebraic closure of E in K is separable over E. Let A,B be E-subspaces of K such that 0

It was conjectured that if n is even, then every permutation of F2n is affine on some 2-dimensional affine subspace of F2n. We prove that the conjecture is true for n = 4, for quadratic permutations of F2n and for permutation polynomials of F2n with coefficients in F2n/2. The conjecture is actually a claim about (AGL(n, 2), AGL(n, 2))-double cosets in permutation group S(F2n) of F2n. We give a formula for the number of (AGL(n, 2), AGL(n, 2))-double cosets in S(F2n) and classify the (AGL(4, 2), AGL(4, 2))-double cosets in S(F24).

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.

Two new classes of permutation polynomials over finite fields are presented: (i) f(x)=(1-x-x^2)x^3^^^e^+^1^2-1-x+x^2 over F"3"^"e where e is a positive even integer; (ii) g"n","p(x)=@?"n"p"=