| 1. |
- Avgustinovich, S. V., et al.
(författare)
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On intersection problem for perfect binary codes
- 2006
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Ingår i: Designs, Codes and Cryptography. - : Springer Science and Business Media LLC. - 0925-1022 .- 1573-7586. ; 39:3, s. 317-322
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Tidskriftsartikel (refereegranskat)abstract
- The main result is that to any even integer q in the interval 0 <= q <= 2(n+1-2) (log(n+1)), there are two perfect codes C-1 and C-2 of length n = 2(m) -1, m >= 4, such that vertical bar C-1 boolean AND C-2 vertical bar = q.
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| 2. |
- Avgustinovich, S. V., et al.
(författare)
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On partitions of an n-cube into nonequivalent perfect codes
- 2007
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Ingår i: Problems of Information Transmission. - : Pleiades Publishing Ltd. - 0032-9460 .- 1608-3253. ; 43:4, s. 310-315
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Tidskriftsartikel (refereegranskat)abstract
- We prove that for all n = 2(k)-1, k >= 5. there exists a partition of the set of all binary vectors of length n into pairwise nonequivalent perfect binary codes of length n with distance 3.
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| 3. |
- Avgustinovich, S. V., et al.
(författare)
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On the structure of symmetry groups of Vasil'ev codes
- 2005
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Ingår i: Problems of Information Transmission. - : Springer Science and Business Media LLC. - 0032-9460 .- 1608-3253. ; 41:2, s. 105-112
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Tidskriftsartikel (refereegranskat)abstract
- The structure of symmetry groups of Vasil'ev codes is studied. It is proved that the symmetry group of an arbitrary perfect binary non-full-rank Vasil'ev code of length n is always nontrivial; for codes of rank n - log(n + 1) + 1, an attainable upper bound on the order of the symmetry group is obtained.
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| 4. |
- Heden, Olof
(författare)
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A full rank perfect code of length 31
- 2006
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Ingår i: Designs, Codes and Cryptography. - : Springer Science and Business Media LLC. - 0925-1022 .- 1573-7586. ; 38:1, s. 125-129
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Tidskriftsartikel (refereegranskat)abstract
- A full rank perfect 1-error correcting binary code of length 31 with a kernel of dimension 21 is described. This was the last open case of the rank-kernel problem of Etzion and Vardy.
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| 5. |
- Heden, Olof
(författare)
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A remark on full rank perfect codes
- 2006
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Ingår i: Discrete Mathematics. - : Elsevier BV. - 0012-365X .- 1872-681X. ; 306:16, s. 1975-1980
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Tidskriftsartikel (refereegranskat)abstract
- Any full rank perfect 1-error correcting binary code of length n = 2(k) - 1 and with a kernel of dimension n - log(n + 1) - m, where in is sufficiently large, may be used to construct a full rank perfect 1-error correcting binary code of length 2(m) - 1 and with a kernel of dimension n - log(n + 1) - k. Especially we may construct full rank perfect 1-error correcting binary codes of length n = 2(m) - 1 and with a kernel of dimension n - log(n + 1) - 4 for nt = 6, 7,..., 10. This result extends known results on the possibilities for the size of a kernel of a full rank perfect code.
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| 6. |
- Heden, Olof
(författare)
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Full rank perfect codes and alpha-kernels
- 2009
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Ingår i: Discrete Mathematics. - : Elsevier BV. - 0012-365X .- 1872-681X. ; 309:8, s. 2202-2216
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Tidskriftsartikel (refereegranskat)abstract
- A perfect 1-error correcting binary code C, perfect code for short, of length n = 2(m) - 1 has full rank if the linear span < C > of the words of C has dimension n as a vector space over the finite field F-2. There are just a few general constructions of full rank perfect codes, that are not given by recursive methods using perfect codes of length shorter than n. In this study we construct full rank perfect codes, the so-called normal alpha-codes, by first finding the superdual of the perfect code. The superdual of a perfect code consists of two matrices G and T in which simplex codes play an important role as subspaces of the row spaces of the matrices G and T. The main idea in our construction is the use of alpha-words. These words have the property that they can be added to certain rows of generator matrices of simplex codes such that the result will be (other) sets of generator matrices for simplex codes. The kernel of these normal alpha-codes will also be considered. It will be proved that any subspace, of the kernel of a normal alpha-code, that satisfies a certain property will be the kernel of another perfect code, of the same length n. In this way, we will be able to relate some of the full rank perfect codes of length n to other full rank perfect codes of the same length n.
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| 7. |
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| 8. |
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| 9. |
- Heden, Olof
(författare)
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On perfect p-ary codes of length p+1
- 2008
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Ingår i: Designs, Codes and Cryptography. - : Springer Science and Business Media LLC. - 0925-1022 .- 1573-7586. ; 46:1, s. 45-56
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Tidskriftsartikel (refereegranskat)abstract
- Let p be a prime number and assume p >= 5. We will use a result of L. Redei to prove, that every perfect 1-error correcting code C of length p + 1 over an alphabet of cardinality p, such that C has a rank equal to p and a kernel of dimension p - 2, will be equivalent to some Hamming code H. Further, C can be obtained from H, by the permutation of the symbols, in just one coordinate position.
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| 10. |
- Heden, Olof, et al.
(författare)
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On the classification of perfect codes : side class structures
- 2006
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Ingår i: Designs, Codes and Cryptography. - : Springer Science and Business Media LLC. - 0925-1022 .- 1573-7586. ; 40:3, s. 319-333
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Tidskriftsartikel (refereegranskat)abstract
- The side class structure of a perfect 1-error correcting binary code (hereafter referred to as a perfect code) C describes the linear relations between the coset representatives of the kernel of C. Two perfect codes C and C' are linearly equivalent if there exists a non-singular matrix A such that AC = C' where C and C' are matrices with the code words of C and C' as columns. Hessler proved that the perfect codes C and C' are linearly equivalent if and only if they have isomorphic side class structures. The aim of this paper is to describe all side class structures. It is shown that the transpose of any side class structure is the dual of a subspace of the kernel of some perfect code and vice versa; any dual of a subspace of a kernel of some perfect code is the transpose of the side class structure of some perfect code. The conclusion is that for classification purposes of perfect codes it is sufficient to find the family of all kernels of perfect codes.
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