Computing Extensions of Linear Codes

This paper deals with the problem of increasing the minimum distance of a linear code by adding one or more columns to the generator matrix. Several methods to compute extensions of linear codes are presented. Many codes improving the previously known lower bounds on the minimum distance have been found.Comment: accepted for publication at ISIT 0

Powerful sets: a generalisation of binary matroids

A set $S\subseteq\{0,1\}^E$ of binary vectors, with positions indexed by $E$, is said to be a \textit{powerful code} if, for all $X\subseteq E$, the number of vectors in $S$ that are zero in the positions indexed by $X$ is a power of 2. By treating binary vectors as characteristic vectors of subsets of $E$, we say that a set $S\subseteq2^E$ of subsets of $E$ is a \textit{powerful set} if the set of characteristic vectors of sets in $S$ is a powerful code. Powerful sets (codes) include cocircuit spaces of binary matroids (equivalently, linear codes over $\mathbb{F}_2$), but much more besides. Our motivation is that, to each powerful set, there is an associated nonnegative-integer-valued rank function (by a construction of Farr), although it does not in general satisfy all the matroid rank axioms. In this paper we investigate the combinatorial properties of powerful sets. We prove fundamental results on special elements (loops, coloops, frames, near-frames, and stars), their associated types of single-element extensions, various ways of combining powerful sets to get new ones, and constructions of nonlinear powerful sets. We show that every powerful set is determined by its clutter of minimal nonzero members. Finally, we show that the number of powerful sets is doubly exponential, and hence that almost all powerful sets are nonlinear.Comment: 19 pages. This work was presented at the 40th Australasian Conference on Combinatorial Mathematics and Combinatorial Computing (40ACCMCC), University of Newcastle, Australia, Dec. 201

A Pseudo DNA Cryptography Method

The DNA cryptography is a new and very promising direction in cryptography research. DNA can be used in cryptography for storing and transmitting the information, as well as for computation. Although in its primitive stage, DNA cryptography is shown to be very effective. Currently, several DNA computing algorithms are proposed for quite some cryptography, cryptanalysis and steganography problems, and they are very powerful in these areas. However, the use of the DNA as a means of cryptography has high tech lab requirements and computational limitations, as well as the labor intensive extrapolation means so far. These make the efficient use of DNA cryptography difficult in the security world now. Therefore, more theoretical analysis should be performed before its real applications. In this project, We do not intended to utilize real DNA to perform the cryptography process; rather, We will introduce a new cryptography method based on central dogma of molecular biology. Since this method simulates some critical processes in central dogma, it is a pseudo DNA cryptography method. The theoretical analysis and experiments show this method to be efficient in computation, storage and transmission; and it is very powerful against certain attacks. Thus, this method can be of many uses in cryptography, such as an enhancement insecurity and speed to the other cryptography methods. There are also extensions and variations to this method, which have enhanced security, effectiveness and applicability.Comment: A small work that quite some people asked abou

Ideal codes over separable ring extensions

This paper investigates the application of the theoretical algebraic notion of a separable ring extension, in the realm of cyclic convolutional codes or, more generally, ideal codes. We work under very mild conditions, that cover all previously known as well as new non trivial examples. It is proved that ideal codes are direct summands as left ideals of the underlying non-commutative algebra, in analogy with cyclic block codes. This implies, in particular, that they are generated by an idempotent element. Hence, by using a suitable separability element, we design an efficient algorithm for computing one of such idempotents

Lattices from Codes for Harnessing Interference: An Overview and Generalizations

In this paper, using compute-and-forward as an example, we provide an overview of constructions of lattices from codes that possess the right algebraic structures for harnessing interference. This includes Construction A, Construction D, and Construction $\pi_A$ (previously called product construction) recently proposed by the authors. We then discuss two generalizations where the first one is a general construction of lattices named Construction $\pi_D$ subsuming the above three constructions as special cases and the second one is to go beyond principal ideal domains and build lattices over algebraic integers