Related papers: A Novel Application of Boolean Functions with High…
Upper bounds on the maximum number of codewords in a binary code of a given length and minimum Hamming distance are considered. New bounds are derived by a combination of linear programming and counting arguments. Some of these bounds…
Most modern block ciphers are built using components whose cryptographic strength is evaluated in terms of their resistance to attacks on the whole cipher. In particular, differential properties of vectorial Boolean functions are studied…
New soft- and hard decision decoding algorithms are presented for general Reed-Muller codes $\left\{\genfrac{}{}{0pt}{}{m}{r}\right\} $ of length $2^{m}$ and distance $2^{m-r}$. We use Plotkin $(u,u+v)$ construction and decompose code…
The binary Reed-Muller codes can be characterized as the radical powers of a modular algebra. We use the Groebner bases to decode these codes.
A notorious open question in circuit complexity is whether Boolean operations of arbitrary arity can efficiently be expressed using modular counting gates only. H{\aa}stad's celebrated switching lemma yields exponential lower bounds for the…
This survey provides a comprehensive overview of the study of the binary and Boolean rank from both a mathematical and a computational perspective, with particular emphasis on their relationship to the real rank. We review the basic…
Let $k\leq n$ be two positive integers and $q$ a prime power. The basic question in minimal linear codes is to determine if there exists an $[n,k]_q$ minimal linear code. The first objective of this paper is to present a new sufficient and…
We present efficient and practical algorithms for a large, distributed system of processors to achieve reliable computations in a secure manner. Specifically, we address the problem of computing a general function of several private inputs…
Boolean networks are dynamical models of disease development in which the activation levels of genes are represented by binary variables. Given a Boolean network, controls represent mutations or medical treatments that fix the activation…
We study a problem of constructing codes that transform a channel with high bit error rate (BER) into one with low BER (at the expense of rate). Our focus is on obtaining codes with smooth ("graceful'') input-output BER curves (as opposed…
Boolean functions are mathematical objects with numerous applications in domains like coding theory, cryptography, and telecommunications. Finding Boolean functions with specific properties is a complex combinatorial optimization problem…
Submodular functions are discrete functions that model laws of diminishing returns and enjoy numerous algorithmic applications. They have been used in many areas, including combinatorial optimization, machine learning, and economics. In…
In this paper we wish to introduce a method to reconstruct large size Welch Bound Equality (WBE) codes from small size WBE codes. The advantage of these codes is that the implementation of ML decoder for the large size codes is reduced to…
BiD codes, which are a new family of algebraic codes of length $3^m$, achieve the erasure channel capacity under bit-MAP decoding and offer asymptotically larger minimum distance than Reed-Muller (RM) codes. In this paper we propose fast…
We study a problem of constructing codes that transform a channel with high bit error rate (BER) into one with low BER (at the expense of rate). Our focus is on obtaining codes with smooth ("graceful") input-output BER curves (as opposed to…
Selman and Kautz's work on ``knowledge compilation'' established how approximation (strengthening and/or weakening) of a propositional knowledge-base can be used to speed up query processing, at the expense of completeness. In this…
Boolean networks constitute relevant mathematical models to study the behaviours of genetic and signalling networks. These networks define regulatory influences between molecular nodes, each being associated to a Boolean variable and a…
Monotone Boolean functions, and the monotone Boolean circuits that compute them, have been intensively studied in complexity theory. In this paper we study the structure of Boolean functions in terms of the minimum number of negations in…
A simple way to generate a Boolean function is to take the sign of a real polynomial in $n$ variables. Such Boolean functions are called polynomial threshold functions. How many low-degree polynomial threshold functions are there? The…
Boolean functions on the space $F_{2}^m$ are not only important in the theory of error-correcting codes, but also in cryptography, where they occur in private key systems. In these two cases, the nonlinearity of these function is a main…