The security of the most popular number-theory public key crypto (PKC) systems will be devastatingly affected by the success of a large quantum computer. Code-based cryptography is one of the promising alternatives that are believed to resist classical and quantum computer attacks. Many families of codes have been proposed for these cryptosystems, one of the main requirements is having an effic...

Our aim is to provide the open domain MATLAB code bvpsuite for the efficient numerical solution of boundary value problems in ordinary differential equations. Motivated by applications, we are especially interested in designing a code whose scope is appropriately wide, including fully implicit problems of mixed orders, parameter dependent problems, problems with unknown parameters, problems pos...

In this paper we study generalized weights as an algebraic invariant of a code. We first describe anticodes in the Hamming and in the rank metric, proving in particular that optimal anticodes in the rank metric coincide with Frobenius-closed spaces. Then we characterize both generalized Hamming and rank weights of a code in terms of the intersection of the code with optimal anticodes in the res...

We prove a conjecture that classifies exceptional numbers. This conjecture arises in two different ways, from cryptography and from coding theory. An odd integer t ≥ 3 is said to be exceptional if f(x) = xt is APN (Almost Perfect Nonlinear) over F2n for infinitely many values of n. Equivalently, t is exceptional if the binary cyclic code of length 2n − 1 with two zeros ω, ωt has minimum distanc...

We introduce the design of a set of code sequences tC n : n ě 1,m ě 1u, with memory order m and code-length N “ Opφq, where φ P p1, 2s is the largest real root of the polynomial equation F pm, ρq “ ρm ́ρm ́1 ́1 and φ is decreasing in m. tC n u is based on the channel polarization idea, where tC n u coincides with the polar codes presented by Arıkan in [1] and can be encoded and decoded with comple...

Tired of writing boilerplate code? Tired of repeating essentially the same function definition for lots of different datatypes? Datatype-generic programming promises to end these coding nightmares. In these lecture notes, we present the key abstractions of datatype-generic programming, give several applications, and provide an elegant embedding of generic programming into Haskell. The embedding...

In classical algebraic coding theory, the minimum distance of block code completely determines the ability of the code in terms of error correction/detection and erasure correction. We have obtained generalizations of these results for network codes.

This report describes a Maple package created for exploring the techniques of Winograd, Nussbaumer, and others for computing “fast” convolution algorithms. After codifying known convolution techniques into a common framework of bilinear algorithms built from parameterized matrices and algebraic operators, Maple’s symbolic and algebraic computation facilities are used to derive and manipulate th...

Generic programming (GP) is a form of abstraction in programming languages that serves to reduce code duplication by exploiting the regular structure of algebraic datatypes. Over the years, several different approaches to GP in Haskell have surfaced. These approaches are often similar, but certain differences make them particularly well-suited for one specific domain or application. As such, th...