AMC
Parity check systems of nonlinear codes over finite commutative Frobenius rings
Thomas Westerbäck

The concept of parity check matrices of linear binary codes has been extended by Heden [10] to parity check systems of nonlinear binary codes. In the present paper we extend this concept to parity check systems of nonlinear codes over finite commutative Frobenius rings. Using parity check systems, results on how to get some fundamental properties of the codes are given. Moreover, parity check systems and its connection to characters is investigated and a MacWilliams type theorem on the distance distribution is given.

keywords: Nonlinear codes finite commutative Frobenius rings parity check systems characters discrete Fourier analysis MacWilliams type theorem
AMC
On the symmetry group of extended perfect binary codes of length $n+1$ and rank $n-\log(n+1)+2$
Olof Heden Fabio Pasticci Thomas Westerbäck
It is proved that for every integer $n=2^k-1$, with $k\geq5$, there exists a perfect code $C$ of length $n$, of rank $r=n-\log(n+1)+2$ and with a trivial symmetry group. This result extends an earlier result by the authors that says that for any length $n=2^k-1$, with $k\geq5$, and any rank $r$, with $n-\log(n+1)+3\leq r\leq n-1$ there exist perfect codes with a trivial symmetry group.
keywords: symmetry group. Perfect codes
AMC
On the existence of extended perfect binary codes with trivial symmetry group
Olof Heden Fabio Pasticci Thomas Westerbäck
The set of permutations of the coordinate set that maps a perfect code $C$ into itself is called the symmetry group of $C$ and is denoted by Sym$(C)$. It is proved that for all integers $n=2^m-1$, where $m=4,5,6,...$, and for any integer $r$, where $n-$log$(n+1)+3\leq r\leq n-1$, there are perfect codes of length $n$ and rank $r$ with a trivial symmetry group, i.e. Sym$(C)=${id}. The result is shown to be true, more generally, for the extended perfect codes of length $n+1$.
keywords: Perfect codes symmetry group.

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