| 1. |
- Heden, Olof, et al.
(författare)
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On linear equivalence and phelps codes
- 2010
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Ingår i: Advances in Mathematics of Communications. - : American Institute of Mathematical Sciences (AIMS). - 1930-5346 .- 1930-5338. ; 4:1, s. 69-81
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 2. |
- Heden, Olof, et al.
(författare)
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On linear equivalence and phelps codes. Addendum
- 2011
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Ingår i: Advances in Mathematics of Communications. - : American Institute of Mathematical Sciences (AIMS). - 1930-5346. ; 5:3, s. 543-546
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 3. |
- 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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| 4. |
- Heden, Olof, et al.
(författare)
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On the classification of perfect codes : Extended side class structures
- 2010
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Ingår i: Discrete Mathematics. - Amsterdam, Netherlands : Elsevier. - 0012-365X .- 1872-681X. ; 310:1, s. 43-55
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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.
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| 5. |
- Hessler, Martin, 1979-
(författare)
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A computer study of some 1-error correcting perfect binary codes
- 2005
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Ingår i: Australasian journal of combinatorics. - 1034-4942. ; 33, s. 217-229
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Tidskriftsartikel (refereegranskat)abstract
- A general algrothm for classifying 1-error correction perfect binary codes of length n, rank n – log2(n+1)+1 and kernel of dimension n – log2/n+1) – 2 is presented. The algorithm gives for n = 31.
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| 6. |
- Hessler, Martin, et al.
(författare)
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Edge cover and polymatroid flow problems
- 2010
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Ingår i: Electronic Journal of Probability. - : Institute of Mathematical Statistics. - 1083-6489. ; 15, s. 2200-2219
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Tidskriftsartikel (refereegranskat)abstract
- In an n by n complete bipartite graph with independent exponentially distributed edge costs, we ask for the minimum total cost of a set of edges of which each vertex is incident to at least one. This so-called minimum edge cover problem is a relaxation of perfect matching. We show that the large n limit cost of the minimum edge cover is W(1)(2) + 2W(1) approximate to 1.456, where W is the Lambert W-function. In particular this means that the minimum edge cover is essentially cheaper than the minimum perfect matching, whose limit cost is pi(2)/6 approximate to 1.645. We obtain this result through a generalization of the perfect matching problem to a setting where we impose a (poly-)matroid structure on the two vertex-sets of the graph, and ask for an edge set of prescribed size connecting independent sets.
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| 7. |
- Hessler, Martin
(författare)
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Perfect codes as isomorphic spaces
- 2006
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Ingår i: Discrete Mathematics. - : Elsevier BV. - 0012-365X .- 1872-681X. ; 306:16, s. 1981-1987
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Tidskriftsartikel (refereegranskat)abstract
- Linear equivalence between perfect codes is defined. This definition gives the concept of general perfect 1-error correcting binary codes. These are defined as 1-error correcting perfect binary codes, with the difference that the set of errors is not the set of weight one words, instead any set with cardinality n and full rank is allowed. The side class structure defines the restrictions on the subspace of any general 1-error correcting perfect binary code. Every linear equivalence class will contain all codes with the same length, rank and dimension of kernel and all codes in the linear equivalence class will have isomorphic side class structures.
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