On self-dual negacirculant codes of index two and four

We study the asymptotic performance of quasi-twisted codes viewed as modules in the ring \(R=\mathbb {F}_q[x]/\langle x^n+1\rangle , \) when they are self-dual and of length 2n or 4n. In particular, in order for the decomposition to be amenable to analysis, we study factorizations of \(x^n+1\) over \(\mathbb {F}_q, \) with n twice an odd prime, containing only three irreducible factors, all self-reciprocal. We give arithmetic conditions bearing on n and q for this to happen. Given a fixed q,  we show these conditions are met for infinitely many n’s, provided a refinement of Artin primitive root conjecture holds. This number theory conjecture is known to hold under generalized Riemann hypothesis (GRH). We derive a modified Varshamov–Gilbert bound on the relative distance of the codes considered, building on exact enumeration results for given n and q.     

Binary extremal self-dual codes of length 60 and related codes

We give a classification of four-circulant singly even self-dual [60, 30, d] codes for \(d=10\) and 12. These codes are used to construct extremal singly even self-dual [60, 30, 12] codes with weight enumerator for which no extremal singly even self-dual code was previously known to exist. From extremal singly even self-dual [60, 30, 12] codes, we also construct optimal singly even self-dual [58, 29, 10] codes with weight enumerator for which no optimal singly even self-dual code was previously known to exist. Finally, we give some restriction on the possible weight enumerators of certain singly even self-dual codes with shadow of minimum weight 1.     

On codes over $$\mathbb {F}_{q}+v\mathbb {F}_{q}+v^{2}\mathbb {F}_{q}$$

In this paper we investigate linear codes with complementary dual (LCD) codes and formally self-dual codes over the ring \(R=\mathbb {F}_{q}+v\mathbb {F}_{q}+v^{2}\mathbb {F}_{q}\), where \(v^{3}=v\), for q odd. We give conditions on the existence of LCD codes and present construction of formally self-dual codes over R. Further, we give bounds on the minimum distance of LCD codes over \(\mathbb {F}_q\) and extend these to codes over R.     

Hadamard full propelinear codes of type Q; rank and kernel

Hadamard full propelinear codes (\(\mathrm{HFP}\)-codes) are introduced and their equivalence with Hadamard groups is proven; on the other hand, the equivalence of Hadamard groups, relative (4n, 2, 4n, 2n)-difference sets in a group, and cocyclic Hadamard matrices, is already known. We compute the available values for the rank and dimension of the kernel of \(\mathrm{HFP}\)-codes of type Q and we show that the dimension of the kernel is always 1 or 2. We also show that when the dimension of the kernel is 2 then the dimension of the kernel of the transposed code is 1 (so, both codes are not equivalent). Finally, we give a construction method such that from an \(\mathrm{HFP}\)-code of length 4n, dimension of the kernel \(k=2\), and maximum rank \(r=2n\), we obtain an \(\mathrm{HFP}\)-code of double length 8n, dimension of the kernel \(k=2\), and maximum rank \(r=4n\).     

A study on monotone self-dual Boolean functions

This paper shows that monotone self-dual Boolean functions in irredundant disjuntive normal form (IDNF) do not have more variables than disjuncts. Monotone self-dual Boolean functions in IDNF with the same number of variables and disjuncts are examined. An algorithm is proposed to test whether a monotone Boolean function in IDNF with n variables and n disjuncts is self-dual. The runtime of the algorithm is O(n³).     

On <Emphasis Type="Italic">G</Emphasis>-covering subgroup systems of finite groups

Let \(\mathcal{F}\) be a class of groups and G a finite group. We call a set Σ of subgroups of G a G-covering subgroup system for  \(\mathcal{F}\) if \(G\in \mathcal{F}\) whenever \(\Sigma \subseteq \mathcal{F}\). Let p be any prime dividing |G| and P a Sylow p-subgroup of G. Then we write Σ _p to denote the set of subgroups of G which contains at least one supplement to G of each maximal subgroup of P. We prove that the sets Σ _p and Σ _p ∪Σ _q , where q≠p, are G-covering subgroup systems for many classes of finite groups.     

Hermitian rank distance codes

Let \(X=X(n,q)\) be the set of \(n\times n\) Hermitian matrices over \(\mathbb {F}_{q^2}\). It is well known that X gives rise to a metric translation association scheme whose classes are induced by the rank metric. We study d-codes in this scheme, namely subsets Y of X with the property that, for all distinct \(A,B\in Y\), the rank of \(A-B\) is at least d. We prove bounds on the size of a d-code and show that, under certain conditions, the inner distribution of a d-code is determined by its parameters. Except if n and d are both even and \(4\le d\le n-2\), constructions of d-codes are given, which are optimal among the d-codes that are subgroups of \((X,+)\). This work complements results previously obtained for several other types of matrices over finite fields.     

Test elements in pro-<Emphasis Type="Italic">p</Emphasis> groups with applications in discrete groups

Let G be a group. An element g ∈ G is called a test element of G if for every endomorphism ? : G → G, ?(g) = g implies ? is an automorphism. We prove that for a finitely generated profinite group G, g ∈ G is a test element of G if and only if it is not contained in a proper retract of G. Using this result we prove that an endomorphism of a free pro-p group of finite rank which preserves an automorphic orbit of a nontrivial element must be an automorphism. We give numerous explicit examples of test elements in free pro-p groups and Demushkin groups. By relating test elements in finitely generated residually finite-p Turner groups to test elements in their pro-p completions, we provide new examples of test elements in free discrete groups and surface groups. Moreover, we prove that the set of test elements of a free discrete group of finite rank is dense in the profinite topology.     

Finite groups with an irreducible character of large degree

Let G be a finite group and d the degree of a complex irreducible character of G, then write |G| = d(d + e) where e is a nonnegative integer. We prove that |G| ≤ e⁴?e³ whenever e > 1. This bound is best possible and improves on several earlier related results.     

On <Emphasis Type="Italic">r</Emphasis>-recognition by prime graph of <Emphasis Type="Italic">B</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>(3) where <Emphasis Type="Italic">p</Emphasis> is an odd prime

Let G be a finite group. The prime graph Γ(G) of G is defined as follows. The vertices of Γ(G) are the primes dividing the order of G and two distinct vertices p and p′ are joined by an edge if there is an element in G of order pp′. We denote by k(Γ(G)) the number of isomorphism classes of finite groups H satisfying Γ(G) = Γ(H). Given a natural number r, a finite group G is called r-recognizable by prime graph if k(Γ(G)) =  r. In Shen et al. (Sib. Math. J. 51(2):244–254, 2010), it is proved that if p is an odd prime, then B _p (3) is recognizable by element orders. In this paper as the main result, we show that if G is a finite group such that Γ(G) = Γ(B _p (3)), where p > 3 is an odd prime, then \({G\cong B_p(3)}\) or C _p (3). Also if Γ(G) = Γ(B ₃(3)), then \({G\cong B_3(3), C_3(3), D_4(3)}\), or \({G/O_2(G)\cong {\rm Aut}(^2B_2(8))}\). As a corollary, the main result of the above paper is obtained.     

Nonlinear functions and difference sets on group actions

There are many generalizations of the classical Boolean bent functions. Let G, H be finite groups and let X be a finite G-set. G-perfect nonlinear functions from X to H have been studied in several papers. They are generalizations of perfect nonlinear functions from G itself to H. By introducing the concept of a (G, H)-related difference family of X, we obtain a characterization of G-perfect nonlinear functions on X in terms of a (G, H)-related difference family. When G is abelian, we prove that there is a normalized G-dual set \(\widehat{X}\) of X, and characterize a G-difference set of X by the Fourier transform on a normalized G-dual set \({{\widehat{X}}}\). We will also investigate the existence and constructions of G-perfect nonlinear functions and G-bent functions. Several known results (IEEE Trans Inf Theory 47(7):2934–2943, 2001; Des Codes Cryptogr 46:83–96, 2008; GESTS Int Trans Comput Sci Eng 12:1–14, 2005; Linear Algebra Appl 452:89–105, 2014) are direct consequences of our results.     

Normal coverings of solvable groups

For a finite non cyclic group G, let γ(G) be the smallest integer k such that G contains k proper subgroups H ₁, . . . , H _k with the property that every element of G is contained in \({H_i^g}\) for some \({i \in \{1,\dots,k\}}\) and \({g \in G.}\) We prove that for every n ≥ 2, there exists a finite solvable group G with γ(G) = n.     

On the rank of compact <Emphasis Type="Italic">p</Emphasis>-adic Lie groups

The rank of a profinite group G is the basic invariant \({{\rm rk}(G):={\rm sup}\{d(H) \mid H \leq G\}}\), where H ranges over all closed subgroups of G and d(H) denotes the minimal cardinality of a topological generating set for H. A compact topological group G admits the structure of a p-adic Lie group if and only if it contains an open pro-p subgroup of finite rank. For every compact p-adic Lie group G one has rk(G) ≥ dim(G), where dim(G) denotes the dimension of G as a p-adic manifold. In this paper we consider the converse problem, bounding rk(G) in terms of dim(G). Every profinite group G of finite rank admits a maximal finite normal subgroup, its periodic radical π(G). One of our main results is the following. Let G be a compact p-adic Lie group such that π(G) = 1, and suppose that p is odd. If \(\{g \in G \mid g^{p-1}=1 \}\) is equal to {1}, then rk(G) = dim(G).     

New characterization of <Emphasis Type="Italic">S</Emphasis><Subscript>4</Subscript>(<Emphasis Type="Italic">q</Emphasis>) by its noncommuting graph

Let G be a nonabelian group, and associate the noncommuting graph ?(G) with G as follows: the vertex set of ?(G) is G\Z(G) with two vertices x and y joined by an edge whenever the commutator of x and y is not the identity. Let S ₄(q) be the projective symplectic simple group, where q is a prime power. We prove that if G is a group with ?(G) ? ?(S ₄(q)) then G ? S ₄(q).     

Riemannian <Emphasis Type="Italic">M</Emphasis>-spaces with homogeneous geodesics

We investigate homogeneous geodesics in a class of homogeneous spaces called M-spaces, which are defined as follows. Let G / K be a generalized flag manifold with \(K=C(S)=S\times K_1\), where S is a torus in a compact simple Lie group G and \(K_1\) is the semisimple part of K. Then, the associated M-space is the homogeneous space \(G/K_1\). These spaces were introduced and studied by H. C. Wang in 1954. We prove that for various classes of M-spaces the only g.o. metric is the standard metric. For other classes of M-spaces we give either necessary, or necessary and sufficient conditions, so that a G-invariant metric on \(G/K_1\) is a g.o. metric. The analysis is based on properties of the isotropy representation \(\mathfrak {m}=\mathfrak {m}_1\oplus \cdots \oplus \mathfrak {m}_s\) of the flag manifold G / K [as \({{\mathrm{Ad}}}(K)\)-modules].     

Almost cover-free codes and designs

An s-subset of codewords of a binary code X is said to be \((s,\,\ell )\) -bad in X if the code X contains a subset of \(\ell \) other codewords such that the conjunction of the \(\ell \) codewords is covered by the disjunctive sum of the s codewords. Otherwise, the s-subset of codewords of X is called \((s,\,\ell )\) -good in X. A binary code X is said to be a cover-free (CF) \((s,\,\ell )\)-code if the code X does not contain \((s,\,\ell )\)-bad subsets. In this paper, we introduce a natural probabilistic generalization of CF \((s,\,\ell )\)-codes, namely: a binary code X is said to be an almost CF \((s,\,\ell )\)-code if the relative number of its \((s,\,\ell )\)-good s-subsets is close to 1. We develop a random coding method based on the ensemble of binary constant weight codes to obtain lower bounds on the capacity of such codes. Our main result shows that the capacity for almost CF \((s,\,\ell )\)-codes is essentially greater than the rate for ordinary CF \((s,\,\ell )\)-codes.     

<Emphasis Type="Italic">s</Emphasis>-Vertex Pancyclic Index

A graph G is vertex pancyclic if for each vertex \({v \in V(G)}\) , and for each integer k with 3 ≤ k ≤ |V(G)|, G has a k-cycle C _k such that \({v \in V(C_k)}\) . Let s ≥ 0 be an integer. If the removal of at most s vertices in G results in a vertex pancyclic graph, we say G is an s-vertex pancyclic graph. Let G be a simple connected graph that is not a path, cycle or K _1,3. Let l(G) = max{m : G has a divalent path of length m that is not both of length 2 and in a K ₃}, where a divalent path in G is a path whose interval vertices have degree two in G. The s-vertex pancyclic index of G, written vp _s (G), is the least nonnegative integer m such that L ^m (G) is s-vertex pancyclic. We show that for a given integer s ≥ 0,

$vp_s(G)\le \left\{\begin{array}{l@{\quad}l}\qquad\quad\quad\,\,\,\,\,\,\, l(G)+s+1: \quad {\rm if} \,\, 0 \le s \le 4 \\ l(G)+\lceil {\rm log}_2(s-2) \rceil+4: \quad {\rm if} \,\, s \ge 5 \end{array}\right.$

And we improve the bound for essentially 3-edge-connected graphs. The lower bound and whether the upper bound is sharp are also discussed.

     

2-Recognizability by prime graph of <Emphasis Type="Italic">PSL</Emphasis>(2, <Emphasis Type="Italic">p</Emphasis><Superscript>2</Superscript>)

Let G be a finite group and let Γ(G) be the prime graph of G. Assume p prime. We determine the finite groups G such that Γ(G) = Γ(PSL(2, p ²)) and prove that if p ≠ 2, 3, 7 is a prime then k(Γ(PSL(2, p ²))) = 2. We infer that if G is a finite group satisfying |G| = |PSL(2, p ²)| and Γ(G) = Γ(PSL(2, p ²)) then G ? PSL(2, p ²). This enables us to give new proofs for some theorems; e.g., a conjecture of W. Shi and J. Bi. Some applications are also considered of this result to the problem of recognition of finite groups by element orders.     

Positive-fraction intersection results and variations of weak epsilon-nets

A connected Finsler space (M, F) is said to be homogeneous if it admits a transitive connected Lie group G of isometries. A geodesic in a homogeneous Finsler space (G / H, F) is called a homogeneous geodesic if it is an orbit of a one-parameter subgroup of G. In this paper, we study the problem of the existence of homogeneous geodesics on a homogeneous Finsler space, and prove that any homogeneous Finsler space of odd dimension admits at least one homogeneous geodesic through each point.     

Induced Cycles in Graphs

The maximum number vertices of a graph G inducing a 2-regular subgraph of G is denoted by \(c_\mathrm{ind}(G)\). We prove that if G is an r-regular graph of order n, then \(c_\mathrm{ind}(G) \ge \frac{n}{2(r-1)} + \frac{1}{(r-1)(r-2)}\) and we prove that if G is a cubic, claw-free graph on order n, then \(c_\mathrm{ind}(G) > \frac{13}{20}n\) and this bound is asymptotically best possible.