Related papers: Computing the Number of Equivalent Classes on $\ma…

Let $R(r,n)$ be the $r$th order Reed-Muller code of length $2^n$. The affine linear group $\text{AGL}(n,\Bbb F_2)$ acts naturally on $R(r,n)$. We derive two formulas concerning the number of orbits of this action: (i) an explicit formula…

Homogeneous rotation symmetric Boolean functions have been extensively studied in recent years because of their applications in cryptography. Little is known about the basic question of when two such functions are affine equivalent. The…

Let $V_n$ be the number of equivalence classes of invertible maps from $\{0,1\}^n$ to $\{0,1\}^n$, under action of permutation of variables on domain and range. So far, the values $V_n$ have been known for $n\le 6$. This paper describes the…

Classification of Non-linear Boolean functions is a long-standing problem in the area of theoretical computer science. In this paper, effort has been made to achieve a systematic classification of all n-variable Boolean functions, where…

We describe the computer-aided classification of equitable partitions of the $12$-cube with quotient matrix $[[2,10],[6,6]]$, or, equivalently, simple orthogonal arrays OA$(1536,12,2,7)$, or order-$7$ correlation-immune Boolean functions in…

This note presents a descending method that allows us to classify quotients of Reed-Muller codes of lenghth 128 under the action of the affine general linear group.

Secure multi-party computation using a physical deck of cards, often called card-based cryptography, has been extensively studied during the past decade. Card-based protocols to compute various Boolean functions have been developed. As each…

This paper concerns the enumeration of isomorphism classes of modules of a polynomial algebra in several variables over a finite field. This is the same as the classification of commuting tuples of matrices over a finite field up to…

Let $n$ be a positive integer, and let $R$ be a (possibly infinite dimensional) finitely presented algebra over a computable field of characteristic zero. We describe an algorithm for deciding (in principle) whether $R$ has at most finitely…

One approach to probabilistic inference involves counting the number of models of a given Boolean formula. Here, we are interested in inferences involving higher-order objects, i.e., functions. We study the following task: Given a Boolean…

Consider the spaces of pseudodifferential operators between tensor density modules over the line as modules of the Lie algebra of vector fields on the line. We compute the equivalence classes of various subquotients of these modules. There…

Monotone Boolean functions (MBFs) are Boolean functions $f: {0,1}^n \rightarrow {0,1}$ satisfying the monotonicity condition $x \leq y \Rightarrow f(x) \leq f(y)$ for any $x,y \in {0,1}^n$. The number of MBFs in n variables is known as the…

Submodular function minimization is a key problem in a wide variety of applications in machine learning, economics, game theory, computer vision, and many others. The general solver has a complexity of $O(n^3 \log^2 n . E +n^4 {\log}^{O(1)}…

We compute the weight distribution of the ${\mathcal R} (4,9)$ by combining the approach described in D. V. Sarwate's Ph.D. thesis from 1973 with knowledge on the affine equivalence classification of Boolean functions. To solve this problem…

Two rational functions $f,g\in\Bbb F_q(X)$ are said to be {\em equivalent} if there exist $\phi,\psi\in\Bbb F_q(X)$ of degree one such that $g=\phi\circ f\circ\psi$. We give an explicit formula for the number of equivalence classes of…

We present a novel class of methods to compute functions of matrices or their action on vectors that are suitable for parallel programming. Solving appropriate simple linear systems of equations in parallel (or computing the inverse of…

In this paper, we approach the study of modules of constant Jordan type and equal images modules over elementary abelian p-groups E_r of rank r \geq 2 by exploiting a functor from the module category of a generalized Beilinson algebra…

We consider a large family of equivalence relations on permutations in Sn that generalise those discovered by Knuth in his study of the Robinson-Schensted correspondence. In our most general setting, two permutations are equivalent if one…

We define a cap in the affine geometry AG(n,2) to be a subset in which every collection of four points is in general position. In this paper, we classify, up to affine equivalence, all caps in AG(7,2) of size k greater than or equal to 10.…

We show that the problem of counting the number of $n$-variable unate functions reduces to the problem of counting the number of $n$-variable monotone functions. Using recently obtained results on $n$-variable monotone functions, we obtain…