The weight distributions of irreducible cyclic codes of length

Let m be a positive integer and q be an odd prime power. In this paper, the weight distributions of all the irreducible cyclic codes of length 2^m over F_q are determined explicitly.     

ACCURATE COMPUTATION FOR IRREDUCIBLE NONSINGULAR M-MATRIX

Computing the eigenvalue of smallest modulus and its corresponding eigneveclor of an irreducible nonsingular M-matrix A is considered, It is shown that if the entries of A are known with high relative accuracy, its eigenvalue of smallest modulus and each component of the corresponding eigenvector will be determined to much higher accuracy than the standard perturbation theory suggests. An algorithm is presented to compute them with a small componentwise backward error, which is consistent with the perturbation results.     

Representations and factors of restriction of scalars

Let L/K be a Galois extension of number fields and let A be an abelian variety defined over K. In this paper we establish the relation between the irreducible characters of the Galois group Gal(L/K) and the simple factors of the restriction of scalars Res_L/K(A) of A from L to K. Then we derive some equivalences of Birch and Swinnerton-Dyer conjectures.     

Recursive constructions of -polynomials over

This paper presents procedures for constructing irreducible polynomials over GF(2^s) with linearly independent roots (or normal polynomials or N-polynomials). For a suitably chosen initial N-polynomial F₀(x)GF(2^s) of degree n, polynomials F_k(x)GF(2^s) of degrees n2^k are constructed by iteratively applying the transformation x→x+x^-1, and their roots are shown to form a normal basis of GF(2^sn2k) over GF(2^s). In addition, the sequences are shown to be trace compatible, i.e., the trace map T_{GF(2^sn2k+1)/GF(2^sn2k)} fromGF(2^sn2k+1) onto GF(2^sn2k) maps the roots of F_k+1(x) onto those of F_k(x).     

Bounds on the kth multi-g base index of nearly reducible sign pattern matrices

Li et al. [On the period and base of a sign pattern matrix, Linear Algebra Appl. 212/213 (1994) 101-120.] extended the concepts of the base and period from nonnegative matrices to powerful sign pattern matrices. Then, Shao and You [Bound on the basis of irreducible generalized sign pattern matrices, Linear Algebra Appl. 427 (2007) 285-300.] extended the concepts of the base from powerful sign pattern matrices to non-powerful irreducible sign pattern matrices. In this paper we mainly study the kth multi-g base index for non-powerful primitive nearly reducible sign pattern matrices. We obtain sharp upper bounds, together with a complete characterization of the equality cases of the kth multi-g base index for primitive nearly reducible generalized sign pattern matrices. We also show that there exist “gaps” in the kth multi-g base index set of the classes of such matrices.     

One-Parameter Irreducible Representation of spl(2,1) Superalgebra

One-parameter irreducible representation of the spl(2,1) superalgebra is studied on subspace and quotient spaces of the universal enveloping algebra of Heisenberg-Weyl superalgebra. The parameter α may be related to the interaction parameter U in one exactly solvable model for correlated electrons. The author was supported financially by Shenzhen science and technology plan project and Shenzhen Institute of Information Technology Grant No. szkj0711.     

A primitive trinomial of degree 6972593

The only primitive trinomials of degree over are and its reciprocal.

     

Corrigenda to ``New primitive whose degree is a Mersenne exponent,' and some new primitive pentanomials

We report an error in our previous paper [#!K1!#], where we announced that we listed all the primitive trinomials over of degree 859433, but there is a bug in the sieve. We missed the primitive trinomial and its reciprocal, as pointed out by Richard Brent et al. We also report some new primitive pentanomials.

     

Irreducibility of unitary group representations and reproducing kernels Hilbert spaces

We discuss unitary representations of groups in Hilbert spaces of functions given together with reproducing kernels, and in particular irreducibility. Our main focus is on examples, including spherical representations of orthogonal groups, distance-transitive finite graphs, irreducibility of induced representations, the discrete series of SL(2, ), and some representations of PGL(2, _p).The appendix contains examples involving groups acting on rooted trees.     

A fast algorithm for testing reducibility of trinomials mod 2 and some new primitive trinomials of degree 3021377

The standard algorithm for testing reducibility of a trinomial of prime degree over requires bits of memory. We describe a new algorithm which requires only bits of memory and significantly fewer memory references and bit-operations than the standard algorithm.

If is a Mersenne prime, then an irreducible trinomial of degree is necessarily primitive. We give primitive trinomials for the Mersenne exponents , , and . The results for extend and correct some computations of Kumada et al. The two results for are primitive trinomials of the highest known degree.