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| 2. |
- Heden, Olof, et al.
(författare)
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ON THE EXISTENCE OF EXTENDED PERFECT BINARY CODES WITH TRIVIAL SYMMETRY GROUP
- 2009
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Ingår i: Advances in Mathematics of Communications. - : American Institute of Mathematical Sciences (AIMS). - 1930-5346 .- 1930-5338. ; 3:3, s. 295-309
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Tidskriftsartikel (refereegranskat)abstract
- 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 <= r <= 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.
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| 3. |
- Pasticci, Fabio, et al.
(författare)
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On rank and kernel of some mixed perfect codes
- 2009
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Ingår i: Discrete Mathematics. - : Elsevier BV. - 0012-365X .- 1872-681X. ; 309:9, s. 2763-2774
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Tidskriftsartikel (refereegranskat)abstract
- Mixed perfect 1-error correcting codes and the associated dual codes over the group Z (n, l), Z (n, l) = under(under(Z2 × Z2 × ⋯ × Z2, {presentation form for vertical right curly bracket}), n) × underover(Z, 2, l), n ≥ 1 and l ≥ 2, are investigated. A lower and an upper bound for the rank k of the kernel of mixed perfect 1-error correcting codes in Z (n, l), depending on the rank r of the mixed perfect code and the structure of the corresponding dual code, are given. Due to a general construction of mixed perfect 1-error correcting group codes in Z (n, l), we show that the upper bound is tight for some n, l and r.
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| 4. |
- Westerbäck, Thomas
(författare)
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Maximal partial packings of Z(2)(n) with perfect codes
- 2007
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Ingår i: Designs, Codes and Cryptography. - : Springer Science and Business Media LLC. - 0925-1022 .- 1573-7586. ; 42:3, s. 335-355
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Tidskriftsartikel (refereegranskat)abstract
- A maximal partial Hamming packing of Z(2)(n) is a family S of mutually disjoint translates of Hamming codes of length n, such that any translate of any Hamming code of length n intersects at least one of the translates of Hamming codes in S. The number of translates of Hamming codes in S is the packing number, and a partial Hamming packing is strictly partial if the family S does not constitute a partition of Z(2)(n). A simple and useful condition describing when two translates of Hamming codes are disjoint or not disjoint is proved. This condition depends on the dual codes of the corresponding Hamming codes. Partly, by using this condition, it is shown that the packing number p, for anymaximal strictly partial Hamming packing of Z(2)(n), n = 2(m)-1, satisfies m + 1 = 4, there exist maximal strictly partial Hamming packings of Z(2)(n) with packing numbers n- 10, n- 9, n- 8,..., n- 1. This implies that the upper bound is tight for any n = 2(m) - 1, m >= 4. All packing numbers for maximal strictly partial Hamming packings of Z(2)(n), n = 7 and 15, are found by a computer search. In the case n = 7 the packing number is 5, and in the case n = 15 the possible packing numbers are 5, 6, 7,..., 13 and 14.
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