On reducibility of n-ary quasigroups

An $n$-ary operation $Q:S^n -> S$ is called an $n$-ary quasigroup of order $|S|$ if in the equation $x_{0}=Q(x_1,...,x_n)$ knowledge of any $n$ elements of $x_0$, ..., $x_n$ uniquely specifies the remaining one. $Q$ is permutably reducible if $Q(x_1,...,x_n)=P(R(x_{s(1)},...,x_{s(k)}),x_{s(k+1)},...,x_{s(n)})$ where $P$ and $R$ are $(n-k+1)$-ary and $k$-ary quasigroups, $s$ is a permutation, and $1<k<n$. An $m$-ary quasigroup $S$ is called a retract of $Q$ if it can be obtained from $Q$ or one of its inverses by fixing $n-m>0$ arguments. We prove that if the maximum arity of a permutably irreducible retract of an $n$-ary quasigroup $Q$ belongs to $\{3,...,n-3\}$, then $Q$ is permutably reducible. Keywords: n-ary quasigroups, retracts, reducibility, distance 2 MDS codes, latin hypercubesComment: 13 pages; presented at ACCT'2004 v2: revised; bibliography updated; 2 appendixe

n-Ary quasigroups of order 4

We characterize the set of all N-ary quasigroups of order 4: every N-ary quasigroup of order 4 is permutably reducible or semilinear. Permutable reducibility means that an N-ary quasigroup can be represented as a composition of K-ary and (N-K+1)-ary quasigroups for some K from 2 to N-1, where the order of arguments in the representation can differ from the original order. The set of semilinear N-ary quasigroups has a characterization in terms of Boolean functions. Keywords: Latin hypercube, n-ary quasigroup, reducibilityComment: 10pp. V2: revise

On the binary codes with parameters of triply-shortened 1-perfect codes

We study properties of binary codes with parameters close to the parameters of 1-perfect codes. An arbitrary binary $(n=2^m-3, 2^{n-m-1}, 4)$ code $C$, i.e., a code with parameters of a triply-shortened extended Hamming code, is a cell of an equitable partition of the $n$-cube into six cells. An arbitrary binary $(n=2^m-4, 2^{n-m}, 3)$ code $D$, i.e., a code with parameters of a triply-shortened Hamming code, is a cell of an equitable family (but not a partition) from six cells. As a corollary, the codes $C$ and $D$ are completely semiregular; i.e., the weight distribution of such a code depends only on the minimal and maximal codeword weights and the code parameters. Moreover, if $D$ is self-complementary, then it is completely regular. As an intermediate result, we prove, in terms of distance distributions, a general criterion for a partition of the vertices of a graph (from rather general class of graphs, including the distance-regular graphs) to be equitable. Keywords: 1-perfect code; triply-shortened 1-perfect code; equitable partition; perfect coloring; weight distribution; distance distributionComment: 12 page

On unbalanced Boolean functions with best correlation immunity

It is known that the order of correlation immunity of a nonconstant unbalanced Boolean function in $n$ variables cannot exceed $2n/3-1$; moreover, it is $2n/3-1$ if and only if the function corresponds to an equitable $2$-partition of the $n$-cube with an eigenvalue $-n/3$ of the quotient matrix. The known series of such functions have proportion $1:3$, $3:5$, or $7:9$ of the number of ones and zeros. We prove that if a nonconstant unbalanced Boolean function attains the correlation-immunity bound and has ratio $C:B$ of the number of ones and zeros, then $CB$ is divisible by $3$. In particular, this proves the nonexistence of equitable partitions for an infinite series of putative quotient matrices. We also establish that there are exactly $2$ equivalence classes of the equitable partitions of the $12$-cube with quotient matrix $[[3,9],[7,5]]$ and $16$ classes, with $[[0,12],[4,8]]$. These parameters correspond to the Boolean functions in $12$ variables with correlation immunity $7$ and proportion $7:9$ and $1:3$, respectively (the case $3:5$ remains unsolved). This also implies the characterization of the orthogonal arrays OA$(1024,12,2,7)$ and OA$(512,11,2,6)$.Comment: v3: final; title changed; revised; OA(512,11,2,6) discusse