| 1. |
- Heden, Olof, et al.
(författare)
-
On linear equivalence and phelps codes
- 2010
-
Ingår i: Advances in Mathematics of Communications. - : American Institute of Mathematical Sciences (AIMS). - 1930-5346 .- 1930-5338. ; 4:1, s. 69-81
-
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.
|
|
| 2. |
- Heden, Olof, et al.
(författare)
-
On linear equivalence and Phelps codes
-
Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- It is shown that all the non full rank FRH-codes, a class of perfect codes we define in the paper, are linearly equivalent to perfect codes obtainable by Phelps construction. It is shown by an example that this class of perfect codes also contains perfect codes that are not obtainable by Phelps construction.
|
|
| 3. |
- Heden, Olof, et al.
(författare)
-
On linear equivalence and phelps codes. Addendum
- 2011
-
Ingår i: Advances in Mathematics of Communications. - : American Institute of Mathematical Sciences (AIMS). - 1930-5346. ; 5:3, s. 543-546
-
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.
|
|
| 4. |
- Heden, Olof, et al.
(författare)
-
On the classification of perfect codes : side class structures
- 2006
-
Ingår i: Designs, Codes and Cryptography. - : Springer Science and Business Media LLC. - 0925-1022 .- 1573-7586. ; 40:3, s. 319-333
-
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.
|
|
| 5. |
- 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.
|
|
| 6. |
- Hessler, Martin, 1979-
(författare)
-
A computer study of some 1-error correcting perfect binary codes
- 2005
-
Ingår i: Australasian journal of combinatorics. - 1034-4942. ; 33, s. 217-229
-
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.
|
|
| 7. |
- Hessler, Martin, 1979-, et al.
(författare)
-
Concentration of the cost of a random matching problem
-
Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- Let Mn be the minimum cost of a perfect matching on a complete graph on n vertices whose edges are assigned independent exponential costs. It follows from work of D. Aldous that Mn converges in probability to π2/12. This was earlier conjectured by M. Mézard and G. Parisi. We establish the more precise result that E | Mn – π2/12| = O(n-1/2).
|
|
| 8. |
- Hessler, Martin, et al.
(författare)
-
Edge cover and polymatroid flow problems
- 2010
-
Ingår i: Electronic Journal of Probability. - : Institute of Mathematical Statistics. - 1083-6489. ; 15, s. 2200-2219
-
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.
|
|
| 9. |
- Hessler, Martin, 1979-, et al.
(författare)
-
LP-relaxed matching with zero-cost loops
-
Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- We study a certain LP-relaxation of the minimum matching problem on a complete graph with random edge costs. In earlier work by Wästlund it was shown that the expected cost of the optimum solution has the simple form 1 – 1/4 + 1/9 – ··· ± 1/n2, an analogue of a corresponding formula for the bipartite problem. Wegeneralize by conditioning on certain edges having zero cost. It is proved that if each node independently is given a zero-cost loop with probability 1 ¡ p then the expected cost of the optimum solution is p – p2/4 + p3/9 – ··· ± pn/n2.
|
|
| 10. |
- Hessler, Martin, 1978-
(författare)
-
Optimization, Matroids and Error-Correcting Codes
- 2009
-
Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- The first subject we investigate in this thesis deals with optimization problems on graphs. The edges are given costs defined by the values of independent exponential random variables. We show how to calculate some or all moments of the distributions of the costs of some optimization problems on graphs.The second subject that we investigate is 1-error correcting perfect binary codes, perfect codes for short. In most work about perfect codes, two codes are considered equivalent if there is an isometric mapping between them. We call this isometric equivalence. Another type of equivalence is given if two codes can be mapped on each other using a non-singular linear map. We call this linear equivalence. A third type of equivalence is given if two codes can be mapped on each other using a composition of an isometric map and a non-singular linear map. We call this extended equivalence.In Paper 1 we give a new better bound on how much the cost of the matching problem with exponential edge costs varies from its mean.In Paper 2 we calculate the expected cost of an LP-relaxed version of the matching problem where some edges are given zero cost. A special case is when the vertices with probability 1 – p have a zero cost loop, for this problem we prove that the expected cost is given by a formula.In Paper 3 we define the polymatroid assignment problem and give a formula for calculating all moments of its cost.In Paper 4 we present a computer enumeration of the 197 isometric equivalence classes of the perfect codes of length 31 of rank 27 and with a kernel of dimension 24.In Paper 5 we investigate when it is possible to map two perfect codes on each other using a non-singular linear map.In Paper 6 we give an invariant for the equivalence classes of all perfect codes of all lengths when linear equivalence is considered.In Paper 7 we give an invariant for the equivalence classes of all perfect codes of all lengths when extended equivalence is considered.In Paper 8 we define a class of perfect codes that we call FRH-codes. It is shown that each FRH-code is linearly equivalent to a so called Phelps code and that this class contains Phelps codes as a proper subset.
|
|
