| 1. |
- Avgustinovich, S. V., et al.
(author)
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On intersection problem for perfect binary codes
- 2006
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In: Designs, Codes and Cryptography. - : Springer Science and Business Media LLC. - 0925-1022 .- 1573-7586. ; 39:3, s. 317-322
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Journal article (peer-reviewed)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.
(author)
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On partitions of an n-cube into nonequivalent perfect codes
- 2007
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In: Problems of Information Transmission. - : Pleiades Publishing Ltd. - 0032-9460 .- 1608-3253. ; 43:4, s. 310-315
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Journal article (peer-reviewed)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.
(author)
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On the structure of symmetry groups of Vasil'ev codes
- 2005
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In: Problems of Information Transmission. - : Springer Science and Business Media LLC. - 0032-9460 .- 1608-3253. ; 41:2, s. 105-112
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Journal article (peer-reviewed)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. |
- Avgustinovich, S. V., et al.
(author)
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The classification of some perfect codes
- 2004
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In: Designs, Codes and Cryptography. - 0925-1022 .- 1573-7586. ; 31:3, s. 313-318
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Journal article (peer-reviewed)abstract
- Perfect 1-error correcting codes C in Z(2)(n), where n = 2(m) - 1, are considered. Let [C] denote the linear span of the words of C and let the rank of C be the dimension of the vector space [C]. It is shown that if the rank of C is n - m + 2 then C is equivalent to a code given by a construction of Phelps. These codes are, in case of rank n - m + 2, described by a Hamming code H and a set of MDS-codes D-h; h is an element of H, over an alphabet with four symbols. The case of rank n - m + 1 is much simpler: Any such code is a Vasil'ev code.
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| 5. |
- El-Zanati, S., et al.
(author)
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Partitions of the 8-Dimensional Vector Space Over GF(2)
- 2010
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In: Journal of combinatorial designs (Print). - : Wiley. - 1063-8539 .- 1520-6610. ; 18:6, s. 462-474
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Journal article (peer-reviewed)abstract
- Let V=V(n,q) denote the vector space of dimension n over GF(q). A set of subspaces of V is called a partition of V if every nonzero vector in V is contained in exactly one subspace of V. Given a. partition P of V with exactly a(i) subspaces of dimension i for 1 <= i <= n, we have Sigma(n)(i=1) a(i)(q(i)-1) = q(n)-1, and we call the n-tuple (a(n), a(n-1), ..., a(1)) the type of P. In this article we identify all 8-tuples (a(8), a(7), ..., a(2), 0) that are the types of partitions of V(8,2).
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| 6. |
- Guzeltepe, Murat, et al.
(author)
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Perfect Mannheim, Lipschitz and Hurwitz weight codes
- 2014
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In: Mathematical Communications. - 1331-0623 .- 1848-8013. ; 19:2, s. 253-276
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Journal article (peer-reviewed)abstract
- The set of residue classes modulo an element pi in the rings of Gaussian integers, Lipschitz integers and Hurwitz integers, respectively, is used as alphabets to form the words of error correcting codes. An error occurs as the addition of an element in a set E to the letter in one of the positions of a word. If epsilon is a group of units in the original rings, then we obtain the Mannheim, Lipschitz and Hurwitz metrics, respectively. Some new perfect 1-error-correcting codes in these metrics are constructed. The existence of perfect 2-error-correcting codes is investigated by computer search.
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| 7. |
- Heden, Olof
(author)
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A full rank perfect code of length 31
- 2006
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In: Designs, Codes and Cryptography. - : Springer Science and Business Media LLC. - 0925-1022 .- 1573-7586. ; 38:1, s. 125-129
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Journal article (peer-reviewed)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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| 8. |
- Heden, Olof
(author)
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A maximal partial spread of size 45 in PG(3,7)
- 2001
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In: Designs, Codes and Cryptography. - 0925-1022 .- 1573-7586. ; 22:3, s. 331-334
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Journal article (peer-reviewed)abstract
- An example of a maximal partial spread of size 45 in PG(3, 7) is given. This example show's that a conjecture of Bruen and Thas from 1976 is false. It also shows that an upper bound for the number of lines of a maximal partial spread, given by Blockhuis in 1994, cannot be improved in general.
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| 9. |
- Heden, Olof
(author)
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A note on the symmetry group of full rank perfect binary codes
- 2012
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In: Discrete Mathematics. - : Elsevier BV. - 0012-365X .- 1872-681X. ; 312:19, s. 2973-2977
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Journal article (peer-reviewed)abstract
- It is proved that the size of the symmetry group Sym(C) of every full rank perfect 1-error correcting binary code C of length n is less than or equal to 2|Sym( Hn)|(n+1), where Hn is a Hamming code of the same length.
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| 10. |
- Heden, Olof
(author)
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A remark on full rank perfect codes
- 2006
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In: Discrete Mathematics. - : Elsevier BV. - 0012-365X .- 1872-681X. ; 306:16, s. 1975-1980
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Journal article (peer-reviewed)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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