Structure of steiner triple systems S(2 m − 1, 3, 2) of rank 2 m − m + 2 over \mathbb{F}_2

The structure of all different Steiner triple systems S(2 ^m ?1, 3, 2) of rank 2 ^m ?m+2 over $\mathbb{F}_2 $ is described. This induces a natural recurrent method for constructing Steiner triple systems of any rank. In particular, the method gives all different such systems of order 2 ^m ? 1 and rank ≤ 2 ^m ? m + 2. The number of such different systems of order 2 ^m ? 1 and rank less than or equal to 2 ^m ? m + 2 which are orthogonal to a given code is found. It is shown that all different triple Steiner systems of order 2 ^m ? 1 and rank ≤ 2 ^m ? m + 2 are derivative and Hamming. Furthermore, all such triples are embedded in quadruple systems of the same rank and in perfect binary nonlinear codes of the same rank.     

Steiner triple systems S(2 m ? 1, 3, 2) of rank 2 m ? m+ 1 over \mathbb{F}_2

Steiner systems S(2 ^m ? 1, 3, 2) of rank 2 ^m ? m+1 over the field $\mathbb{F}_2$ are considered. A new recursive method for constructing Steiner triple systems of an arbitrary rank is proposed. The number of all Steiner systems of rank 2 ^m ? m+1 is obtained. Moreover, it is shown that all Steiner triple systems S(2 ^m ? 1, 3, 2) of rank r ?? 2 ^m ? m+1 are derived, i.e., can be completed to Steiner quadruple systems S(2 ^m , 4, 3). It is also proved that all such Steiner triple systems are Hamming; i.e., any Steiner triple system S(2 ^m ? 1, 3, 2) of rank r ?? 2 ^m ? m + 1 over the field $\mathbb{F}_2$ occurs as the set of words of weight 3 of a binary nonlinear perfect code of length 2 ^m ?1.     

Involutive method for computing Gröbner bases over $$ \mathbb{F}_2 $$

In this paper, an involutive algorithm for computation of Gröbner bases for polynomial ideals in a ring of polynomials in many variables over the finite field \(\mathbb{F}_2 \) with the values of variables belonging of \(\mathbb{F}_2 \) is considered. The algorithm uses Janet division and is specialized for a graded reverse lexicographical order of monomials. We compare efficiency of this algorithm and its implementation in C++ with that of the Buchberger algorithm, as well as with the algorithms of computation of Gröbner bases that are built in the computer algebra systems Singular and CoCoA and in the FGb library for Maple. For the sake of comparison, we took widely used examples of computation of Gröbner bases over ? and adapted them for \(\mathbb{F}_2 \). Polynomial systems over \(\mathbb{F}_2 \) with the values of variables in \(\mathbb{F}_2 \) are of interest, in particular, for modeling quantum computation and a number of cryptanalysis problems.     

Fast computation of Gröbner basis of homogenous ideals of $$ \mathbb{F} $$[ x,y]

This paper provides a fast algorithm for Grobnerbases of homogenous ideals of F[x, y] over a finite field F. We show that only the 8-polynomials of neighbor pairs of a strictly ordered finite homogenours generating set are needed in the computing of a Grobner base of the homogenous ideal. It reduces dramatically the number of unnecessary 5-polynomials that are processed. We also show that the computational complexity of our new algorithm is O（N^2）, where N is the maximum degree of the input generating polynomials. The new algorithm can be used to solve a problem of blind recognition of convolutional codes. This problem is a new generalization of the important problem of synthesis of a linear recurring sequence.     

Vasil’ev codes of length n = 2m and doubling of Steiner systems S(n, 4, 3) of a given rank

Extended binary perfect nonlinear Vasil’ev codes of length n = 2^m and Steiner systems S(n, 4, 3) of rank n-m over F ₂ are studied. The generalized concatenated construction of Vasil’ev codes induces a variant of the doubling construction for Steiner systems S(n, 4, 3) of an arbitrary rank r over F ₂. We prove that any Steiner system S(n = 2^m, 4, 3) of rank n-m can be obtained by this doubling construction and is formed by codewords of weight 4 of these Vasil’ev codes. The length 16 is studied in detail. Orders of the full automorphism groups of all 12 nonequivalent Vasil’ev codes of length 16 are found. There are exactly 15 nonisomorphic systems S(16, 4, 3) of rank 12 over F ₂, and they can be obtained from codewords of weight 4 of the extended Vasil’ev codes. Orders of the automorphism groups of all these Steiner systems are found.     

On the Comparison of Proof Planning Systems: , Ωmega and IsaPlanner

We present a framework for describing proof planners. This framework is based around a decomposition of proof planners into planning states, proof language, proof plans, proof methods, proof revision, proof control and planning algorithms.We use this framework to motivate the comparison of three recent proof planning systems, λCLaM, Ωmega and IsaPlanner, and demonstrate how the framework allows us to discuss and illustrate both their similarities and differences in a consistent fashion. This analysis reveals that proof control and the use of contextual information in planning states are key areas in need of further investigation.