AMC
The partial order of perfect codes associated to a perfect code
Olof Heden
Advances in Mathematics of Communications 2007, 1(4): 399-412 doi: 10.3934/amc.2007.1.399
It is clarified whether or not ''full rank perfect 1-error correcting binary codes act like primes in the family of all perfect 1-error correcting binary codes''. Thereby the well known connection between perfect 1-error correcting binary codes and tilings will be discussed and used.
keywords: Perfect codes tiles.
AMC
Partitions of $\mathbb F$n into non-parallel Hamming codes
Olof Heden Faina I. Solov’eva
Advances in Mathematics of Communications 2009, 3(4): 385-397 doi: 10.3934/amc.2009.3.385
We investigate partitions of the set $\mathbb F$n of all binary vectors of length $n$ into cosets of pairwise distinct linear Hamming codes (''non-parallel Hamming codes'') of length $n$. We present several constructions of partitions of $\mathbb F$n into non-parallel Hamming codes of length $n$ and discuss a lower bound on the number of different such partitions.
keywords: Phelps partition. Perfect code Hamming code non-parallel Hamming codes
AMC
On the structure of non-full-rank perfect $q$-ary codes
Olof Heden Denis S. Krotov
Advances in Mathematics of Communications 2011, 5(2): 149-156 doi: 10.3934/amc.2011.5.149
The Krotov combining construction of perfect $1$-error-correcting binary codes from 2000 and a theorem of Heden saying that every non-full-rank perfect $1$-error-correcting binary code can be constructed by this combining construction is generalized to the $q$-ary case. Simply speaking, every non-full-rank perfect code $C$ is the union of a well-defined family of $\bar\mu$-components K$\bar\mu$, where $\bar\mu$ belongs to an “outer” perfect code C*, and these components are at distance three from each other. Components from distinct codes can thus freely be combined to obtain new perfect codes. The Phelps general product construction of perfect binary code from 1984 is generalized to obtain $\bar\mu$-components, and new lower bounds on the number of perfect $1$-error-correcting $q$-ary codes are presented.
keywords: lower bound. $q$-ary codes Perfect codes components
AMC
A survey of perfect codes
Olof Heden
Advances in Mathematics of Communications 2008, 2(2): 223-247 doi: 10.3934/amc.2008.2.223
The first examples of perfect $e$-error correcting $q$-ary codes were given in the 1940's by Hamming and Golay. In 1973 Tietäväinen, and independently Zinoviev and Leontiev, proved that if q is a power of a prime number then there are no unknown multiple error correcting perfect $q$-ary codes. The case of single error correcting perfect codes is quite different. The number of different such codes is very large and the classification, enumeration and description of all perfect 1-error correcting codes is still an open problem.
   This survey paper is devoted to the rather many recent results, that have appeared during the last ten years, on perfect 1-error correcting binary codes. The following topics are considered: Constructions, connections with tilings of groups and with Steiner Triple Systems, enumeration, classification by rank and kernel dimension and by linear equivalence, reconstructions, isometric properties and the automorphism group of perfect codes.
keywords: Perfect codes.
AMC
On linear equivalence and Phelps codes. Addendum
Olof Heden Martin Hessler
Advances in Mathematics of Communications 2011, 5(3): 543-546 doi: 10.3934/amc.2011.5.543
A new class of perfect 1-error correcting binary codes, so called RRH-codes, are identified, and it is shown that every such code is linearly equivalent to a perfect code obtainable by the Phelps construction.
keywords: Phelps codes Perfect codes tilings.
AMC
On linear equivalence and Phelps codes
Olof Heden Martin Hessler
Advances in Mathematics of Communications 2010, 4(1): 69-81 doi: 10.3934/amc.2010.4.69
It is shown that all non-full-rank FRH-codes, a class of perfect codes we define in this paper, are linearly equivalent to perfect codes obtainable by Phelps' construction. Moreover, it is shown by an example that the class of perfect FRH-codes also contains perfect codes that are not obtainable by Phelps construction.
keywords: tilings. Perfect codes
AMC
On the existence of extended perfect binary codes with trivial symmetry group
Olof Heden Fabio Pasticci Thomas Westerbäck
Advances in Mathematics of Communications 2009, 3(3): 295-309 doi: 10.3934/amc.2009.3.295
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.
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
Advances in Mathematics of Communications 2012, 6(2): 121-130 doi: 10.3934/amc.2012.6.121
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

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