| 1. |
- Britz, Thomas, et al.
(författare)
-
Demi-matroids from codes over finite Frobenius rings
- 2015
-
Ingår i: Designs, Codes and Cryptography. - : Springer Science and Business Media LLC. - 0925-1022 .- 1573-7586. ; 75:1, s. 97-107
-
Tidskriftsartikel (refereegranskat)abstract
- We present a construction of demi-matroids, a generalization of matroids, from linear codes over finite Frobenius rings, as well as a Greene-type identity for rank generating functions of demi-matroids. We also prove a MacWilliams-type identity for Hamming support enumerators of linear codes over finite Frobenius rings. As a special case, these results give a combinatorial proof of the MacWilliams identity for Hamming weight enumerators of linear codes over finite Frobenius rings.
|
|
| 2. |
|
|
| 3. |
|
|
| 4. |
- Heden, Olof, et al.
(författare)
-
On the classification of perfect codes : Extended side class structures
- 2010
-
Ingår i: Discrete Mathematics. - Amsterdam, Netherlands : Elsevier. - 0012-365X .- 1872-681X. ; 310:1, s. 43-55
-
Tidskriftsartikel (refereegranskat)abstract
- The two 1-error correcting perfect binary codes, C and C′ are said to be equivalent if there exists a permutation π of the set of the n coordinate positions and a word such that . Hessler defined C and C′ to be linearly equivalent if there exists a non-singular linear map φ such that C′=φ(C). Two perfect codes C and C′ of length n will be defined to be extended equivalent if there exists a non-singular linear map φ and a word such thatHeden and Hessler, associated with each linear equivalence class an invariant LC and this invariant was shown to be a subspace of the kernel of some perfect code. It is shown here that, in the case of extended equivalence, the corresponding invariant will be the extension of the code LC.This fact will be used to give, in some particular cases, a complete enumeration of all extended equivalence classes of perfect codes.
|
|
| 5. |
- Heden, Olof, et al.
(författare)
-
ON THE EXISTENCE OF EXTENDED PERFECT BINARY CODES WITH TRIVIAL SYMMETRY GROUP
- 2009
-
Ingår i: Advances in Mathematics of Communications. - : American Institute of Mathematical Sciences (AIMS). - 1930-5346 .- 1930-5338. ; 3:3, s. 295-309
-
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.
|
|
| 6. |
- Heden, Olof, et al.
(författare)
-
on the symmetry group of extended perfect binary codes of length n+1 and rank n-log(n+1)+2
- 2012
-
Ingår i: Advances in Mathematics of Communication. - : American Institute of Mathematical Sciences (AIMS). - 1930-5346 .- 1930-5338. ; 6:2, s. 121-130
-
Tidskriftsartikel (refereegranskat)abstract
- It is proved that for every integer n = 2(k) - 1, with k >= 5, 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 >= 5, and any rank r, with n - log(n + 1) + 3 <= r <= n - 1 there exist perfect codes with a trivial symmetry group.
|
|
| 7. |
- Kaski, Petteri, et al.
(författare)
-
Fast Möbius inversion in semimodular lattices and ER-labelable posets
- 2016
-
Ingår i: The Electronic Journal of Combinatorics. - : The Electronic Journal of Combinatorics. - 1097-1440 .- 1077-8926. ; 23:3, s. 1-13
-
Tidskriftsartikel (refereegranskat)abstract
- We consider the problem of fast zeta and Möbius transforms in finite posets, particularly in lattices. It has previously been shown that for a certain family of lattices, zeta and Möbius transforms can be computed in $O(e)$ elementary arithmetic operations, where $e$ denotes the size of the covering relation. We show that this family is exactly that of geometric lattices. We also extend the algorithms so that they work in $e$ operations for all semimodular lattices, including chains and divisor lattices. Finally, for both transforms, we provide a more general algorithm that works in $e$ operations for all ER-labelable posets.
|
|
| 8. |
- Pasticci, Fabio, et al.
(författare)
-
On rank and kernel of some mixed perfect codes
- 2009
-
Ingår i: Discrete Mathematics. - : Elsevier BV. - 0012-365X .- 1872-681X. ; 309:9, s. 2763-2774
-
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.
|
|
| 9. |
- Westerbäck, Thomas
(författare)
-
Maximal partial packings of Z(2)(n) with perfect codes
- 2007
-
Ingår i: Designs, Codes and Cryptography. - : Springer Science and Business Media LLC. - 0925-1022 .- 1573-7586. ; 42:3, s. 335-355
-
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.
|
|
| 10. |
|
|
