Pavel A. Panteleev

Preset Distinguishing Sequences and Diameter of Transformation Semigroups

We investigate the length \(\ell (n,k)\) of a shortest preset distinguishing sequence (PDS) in the worst case for a \(k\)-element subset of an \(n\)-state Mealy automaton. It was mentioned by Sokolovskii [18] that this problem is closely related to the problem of finding the maximal subsemigroup diameter \(\ell (\mathbf {T}_n)\) for the full transformation semigroup \(\mathbf {T}_n\) of an \(n\)-element set. We prove that \(\ell (\mathbf {T}_n)=2^n\exp \{\sqrt{\frac{n}{2}\ln n}(1+ o(1))\}\) as \(n\rightarrow \infty \) and, using approach of Sokolovskii, find the asymptotics of \(\log _2 \ell (n,k)\) as \(n,k\rightarrow \infty \) and \(k/n\rightarrow a\in (0,1)\).

Finite automata and numbers

Abstract We study finite automata representations of numerical rings. Such representations correspond to the class of linear p-adic automata that compute homogeneous linear functions with rational coefficients in the ring of p-adic integers. Finite automata act both as ring elements and as operations. We also study properties of transition diagrams of automata that compute a function f(x)= cx of one variable. In particular we obtain precise values for the number of states of such automata and show that for c > 0 transition diagrams are self-dual (this property generalises self-duality of Boolean functions). We also obtain the criterion for an automaton computing a function f(x)= cx to be a permutation automaton, and fully describe groups that are transition semigroups of such automata.

Fast systematic encoding of quasi-cyclic codes using the Chinese remainder theorem

Quasi-cyclic (QC) codes are a wide class of error-correcting codes possessing nice theoretical properties and having many practical applications. This paper provides a new approach to the problem of efficient encoding of QC codes based on the Chinese remainder theorem (CRT). We present a fast systematic CRT-based encoding algorithm that has superior asymptotic complexity than the previous methods based on shift registers. We also consider the encoding problem for QC low-density parity-check (LDPC) codes. In the special case when the parity part of a sparse parity-check QC matrix has a QC generalized inverse we propose a systematic CRT-based encoding algorithm that can exploit the parity-check matrix sparseness. We also give necessary and sufficient conditions when a QC matrix over an arbitrary field has a QC generalized inverse of the same circulant size.

On distinguishability of states of automata

We investigate variants of the notion of distinguishability of automata. The distinguishability in the sense of a given metric on the set of output symbols, the k-distinguishability and the ∞-distinguishability are considered. For each variant the exact value of the corresponding Shannon function is obtained.We find the minimum value of the parameter k for which the k-distinguishability implies the ∞-distinguishability.