SYM-ILDL is a numerical software package that computes incomplete LDLT (or ‘ILDL’) factorizations of symmetric indefinite and skew-symmetric matrices. The core of the algorithm is a Crout variant of incomplete LU (ILU), originally introduced and implemented for symmetric matrices by [Li and Saad, Crout versions of ILU factorization with pivoting for sparse symmetric matrices, Transactions on Nu...

We investigate various aspects of involutions of groups, i.e, anti-automorphisms of order at most two. The emphasis is on finite abelian groups. We count the number of involutions for the cyclic groups, and consider the problem for direct products of groups. We also give a characterization for the set of skewed squares of finitely generated abelian groups with identity as the involution. The pr...

We investigate various aspects of involutions of groups, i.e, anti-automorphisms of order at most two. The emphasis is on finite abelian groups. We count the number of involutions for the cyclic groups, and consider the problem for direct products of groups. We also give a characterization for the set of skewed squares of finitely generated abelian groups with identity as the involution. The pr...

In this paper, we define and study quantum cyclic codes, a generalisation of cyclic codes to the quantum setting. Previously studied examples of quantum cyclic codes were all quantum codes obtained from classical cyclic codes via the CSS construction. However, the codes that we study are much more general. In particular, we construct cyclic stabiliser codes with parameters [[5, 1, 3]], [[17, 1,...

A general theorem is proved showing how to ohtain a constant-weight binary cyclic code from a p-ary linear cyclic code, where p is a prime,. by using a representation of Cl;(p) as cyclic shifts of a binary p-tuple. Based on this theorem, constructions are given for four classes of binary constant-weight codes. The first two classes are shown to achieve the Johnson upper bound on minimum distanc...

A Z2Z4-additive code C ⊆ Z α 2 ×Z 4 is called cyclic if the set of coordinates can be partitioned into two subsets, the set of Z2 and the set of Z4 coordinates, such that any cyclic shift of the coordinates of both subsets leaves the code invariant. Let Φ(C) be the binary Gray image of C. We study the rank and the dimension of the kernel of a Z2Z4-additive cyclic code C, that is, the dimensions...

A cyclic code is associated with another cyclic code to bound its minimum distance. The algebraic relation between these two codes allows the formulation of syndromes and a key equation. In this contribution, we outline the decoding approach for the case of errors and erasures and show how the Extended Euclidean Algorithm can be used for decoding.

N MOST memory devices, information is stored in two I dimensions. In such cases, errors usually take the form of two-dimensional bursts. For example, VLSI RAM chips are sensitive to alpha particles and other radiation effects which can cause two-dimensional bursts of errors. Twodimensional burst correcting codes can be used to combat such errors. Elspas [ l ] has shown that the product of two c...

A new bound on the minimum distance of q-ary cyclic codes is proposed. It is based on the description by another cyclic code with small minimum distance. The connection to the BCH bound and the Hartmann–Tzeng (HT) bound is formulated explicitly. We show that for many cases our approach improves the HT bound. Furthermore, we refine our bound for several families of cyclic codes. We define syndro...