Volume 14, Issue 2

It is known that the algebra of linear recurring sequences over a commutative ring R is the Hopf algebra dual to the polynomial algebra over R . In this paper, we consider some concepts and operations of the theory of Hopf algebras and modules which have interesting interpretations in terms of linear recurring sequences.

Necessary conditions for solvability of systems of linear equations over associative rings are considered. In some cases, these conditions are also sufficient for solvability of systems.

We give a complete description (with the use of computation) of the best parameters of linear codes that correspond to the left ideals in the loop algebras F q L for q ∈ {2,3,4,5} and | L | ≤ 7, and also in the group algebras F q G for groups G of order | G | ≤ 12. We distinguish the linearly optimal codes, the codes satisfying the Varshamov–Hilbert condition as well as those for which the Plotkin bound is attained. The results suggest that the research in codes constructed by using non-associative and non-semisimple non-commutative algebras can open new possibilities and deserves to be developed.

We show that the complexity of a Zhegalkin polynomial of degree k, 2 ≤ k ≤ n , does not exceed (as n → ∞) and asymptotically coincides with this number for almost all such polynomials. We also show that the complexity of any homogeneous Zhegalkin polynomial of degree k (for most values of k ) does not exceed (as n → ∞) and asymptotically coincides with this number for almost all such polynomials.

In the study of the problem of existence and conjugacy in an arbitrary finite group it is known the Blessenohl-Laue result that in any finite group G there exists a unique class of conjugate quasinilpotent injectors which are exactly the ℜ ∗ -maximal subgroups of G containing the generalised Fitting subgroup F ∗ ( G ). In this paper, with the use of constructions of the Blessenohl–Laue and Gaschütz classes, we extend the Blessenohl-Laue result to the case of the Fitting class 𐔉 = 𐕳𐔅, where 𐕳 is a non-empty Fitting class and 𐔅 is a Blessenohl-Laue class, and thus we distinguish a new class of conjugate 𐔉-injectors in the classes 𐔈 of all finite groups and 𐔖 π of all finite π -solvable groups respectively. Moreover, we prove that the 𐔉-injectors of the group G are exactly all 𐔉-maximal subgroups of G , which contain its 𐔉-radical G 𐔉. Special cases of such injectors are the injectors for many known Fitting classes. In particular, such injectors in the class 𐔖 of all finite solvable groups were described by B. Hartley, B. Fischer, W. Frantz, and P. Lockett.

Let Γ be a strongly regular graph with parameters ( v , k , 1, 2). Then k = u 2 + u + 2 and u = 1, 3, 4, 10, or 31. It is known that such graphs exist for u equal to 1 and 4. They are the (3 × 3)-lattice and the graph of cosets of the ternary Golay code. If u = 3, then Γ has the parameters (99, 14, 1, 2). The question on existence of such graphs was posed by J. Seidel. With the use of theory of characters of finite groups we find the possible orders and the structures of subgraphs of the fixed points of automorphisms of the graph Γ with parameters (99, 14, 1, 2). It is proved that if the group Aut(Γ) contains an involution, then its order divides 42.

We find the asymptotic behaviour of the probability of large deviations P ( W L, L ≥ θL ) of the Shepp statistic W L, L which is equal to the maximum of fluctuations of the random walk in the window of width L moving in the interval [1, 2 L ] as L → ∞ and θ is a constant. We assume that ξ 1 , ξ 2 , ... are independent identically distributed random variables with non-lattice distribution satisfying the right-side Cramer condition. We show that the asymptotics are of the form H θ L P ( S l ≥ θL ), where Hθ is a constant depending on θ .