Perfect codes from PGL(2,5) in Star graphs

The Star graph $S_n$ is the Cayley graph of the symmetric group $Sym_n$ with the generating set \{(1\mbox{ }i): 2\leq i\leq n \}. Arumugam and Kala proved that $\{\pi\in Sym_n: \pi(1)=1\}$ is a perfect code in $S_n$ for any $n, n\geq 3$. In this note we show that for any $n, n\geq 6$ the Star graph $S_n$ contains a perfect code which is a union of cosets of the embedding of $PGL(2,5)$ into $Sym_6$

Equitable 2-partitions of the Hamming graphs with the second eigenvalue

The eigenvalues of the Hamming graph $H(n,q)$ are known to be $\lambda_i(n,q)=(q-1)n-qi$, $0\leq i \leq n$. The characterization of equitable 2-partitions of the Hamming graphs $H(n,q)$ with eigenvalue $\lambda_{1}(n,q)$ was obtained by Meyerowitz in [15]. We study the equitable 2-partitions of $H(n,q)$ with eigenvalue $\lambda_{2}(n,q)$. We show that these partitions are reduced to equitable 2-partitions of $H(3,q)$ with eigenvalue $\lambda_{2}(3,q)$ with exception of two constructions

A concatenation construction for propelinear perfect codes from regular subgroups of GA(r,2)

A code $C$ is called propelinear if there is a subgroup of $Aut(C)$ of order $|C|$ acting transitively on the codewords of $C$. In the paper new propelinear perfect binary codes of any admissible length more than $7$ are obtained by a particular case of the Solov'eva concatenation construction--1981 and the regular subgroups of the general affine group of the vector space over $GF(2)$

A note on Cameron - Liebler line classes in PG(n,4)

A {\it Cameron -- Liebler line class} ${\cal L}$ with parameter $x$ is a set of lines of projective geometry $PG(3,q)$ such that each line of ${\cal L}$ meets exactly $x(q+1)+q^2-1$ lines of ${\cal L}$ and each line that is not from ${\cal L}$ meets exactly $x(q+1)$ lines of ${\cal L}$. In this paper, we obtain a classification of Cameron -- Liebler line classes in PG(3,4) and a classification of their generalization in PG(n,4), $n\geqslant 4$

On components of a Kerdock code and the dual of the BCH code C1,3C_{1,3}

In the paper we investigate the structure of $i$-components of two classes of codes: Kerdock codes and the duals of the primitive cyclic BCH code with designed distance 5 of length $n=2^m-1$, for odd $m$. We prove that for any admissible length a punctured Kerdock code consists of two $i$-components and the dual of BCH code is a $i$-component for any $i$. We give an alternative proof for the fact presented by De Caen and van Dam in 1999 that the restriction of the Hamming scheme to a doubly shortened Kerdock code is an association scheme

On symmetry group of Mollard code

For a pair of given binary perfect codes C and D of lengths t and m respectively, the Mollard construction outputs a perfect code M(C,D) of length tm + t + m, having subcodes C1 and D2, that are obtained from codewords of C and D respectively by adding appropriate number of zeros. In this work we generalize of a result for symmetry groups of Vasilev codes [2] and find the group Stab_{D2}Sym(M(C,D)). The result is preceded by and partially based on a discussion of linearity of coordinate positions (points) in a nonlinear perfect code (non-projective Steiner triple system respectively).Comment: submitted to Electronic Journal of Combinatoric

Minimum supports of eigenfunctions of Johnson graphs

We study the weights of eigenvectors of the Johnson graphs $J(n,w)$. For any $i \in \{1,\ldots,w\}$ and sufficiently large $n, n\geq n(i,w)$ we show that an eigenvector of $J(n,w)$ with the eigenvalue $\lambda_i=(n-w-i)(w-i)-i$ has at least $2^i(^{n-2i}_{w-i})$ nonzeros and obtain a characterization of eigenvectors that attain the bound

On weak isometries of Preparata codes

Let C1 and C2 be codes with code distance d. Codes C1 and C2 are called weakly isometric, if there exists a mapping J:C1->C2, such that for any x,y from C1 the equality d(x,y)=d holds if and only if d(J(x),J(y))=d. Obviously two codes are weakly isometric if and only if the minimal distance graphs of these codes are isomorphic. In this paper we prove that Preparata codes of length n>=2^12 are weakly isometric if and only if these codes are equivalent. The analogous result is obtained for punctured Preparata codes of length not less than 2^10-1.Comment: Submitted to Problems of Information Transmission on 11th of January 200

MMS-type problems for Johnson scheme

In the current work we consider the minimization problems for the number of nonzero or negative values of vectors from the first and second eigenspaces of the Johnson scheme respectively. The topic is a meeting point for generalizations of the Manikam-Mikl\'{o}s-Singhi conjecture proven by Blinovski and the minimum support problem for the eigenspaces of the Johnson graph, asymptotically solved by authors in a recent paper.Comment: 9 page

Ranks of propelinear perfect binary codes

It is proven that for any numbers n=2^m-1, m >= 4 and r, such that n - log(n+1)<= r <= n excluding n = r = 63, n = 127, r in {126,127} and n = r = 2047 there exists a propelinear perfect binary code of length n and rank r