A q-analogous of the characterization of hypercubes as graphs

Ceccherini, Pier Vittorio

doi:10.1007/bf01230125

Cited by 11 publications

(7 citation statements)

References 2 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Remark: The fact that d(A, B) is a metric also follows from the fact that this quantity represents the distance of a geodesic between A and B in the undirected Hasse graph representing the lattice of subspaces of W [11] partially ordered by inclusion, i.e., where X Y if and only if X is a subspace of Y . In this graph, the vertices correspond to the elements of P(W ) and an edge joins a subspace X with a subspace Y if and only if | dim(X) − dim(Y )| = 1 and either X ⊂ Y or Y ⊂ X.…”

Section: A Metric On P(w )mentioning

confidence: 99%

“…We also note that the distance between two spaces U, V in P(W, ℓ) introduced in (3) is, like in the case of constant-weight codes in the Hamming metric, an even number equal to twice the graph distance in the Grassmann graph. 2 As noted Section 1, codes in the Grassmann graph have been considered previously in [11][12][13][14][15]17].…”

Section: Constant-dimension Codesmentioning

confidence: 99%

See 1 more Smart Citation

Coding for Errors and Erasures in Random Network Coding

Koetter

Kschischang

2007

2007 IEEE International Symposium on Information Theory

426

1,175

View full text Add to dashboard Cite

The problem of error-control in random linear network coding is considered. A "noncoherent" or "channel oblivious" model is assumed where neither transmitter nor receiver is assumed to have knowledge of the channel transfer characteristic. Motivated by the property that linear network coding is vector-space preserving, information transmission is modelled as the injection into the network of a basis for a vector space V and the collection by the receiver of a basis for a vector space U . A metric on the projective geometry associated with the packet space is introduced, and it is shown that a minimum distance decoder for this metric achieves correct decoding if the dimension of the space V ∩U is sufficiently large. If the dimension of each codeword is restricted to a fixed integer, the code forms a subset of a finite-field Grassmannian, or, equivalently, a subset of the vertices of the corresponding Grassmann graph. Sphere-packing and sphere-covering bounds as well as a generalization of the Singleton bound are provided for such codes. Finally, a Reed-Solomon-like code construction, related to Gabidulin's construction of maximum rank-distance codes, is described and a Sudan-style "list-1" minimum distance decoding algorithm is provided. * Supported in part by DARPA ITMANET W911NF-07-I-0029.

show abstract

Section: A Metric On P(w )mentioning

confidence: 99%

Section: Constant-dimension Codesmentioning

confidence: 99%

Coding for Errors and Erasures in Random Network Coding

Koetter

Kschischang

2007

2007 IEEE International Symposium on Information Theory

426

1,175

View full text Add to dashboard Cite

show abstract

“…Now the bulk corresponds to the Brody-Hughston quantum space-time structure 23 equipped with the chronometric form of Eq. (82).…”

Section: Summary Of Resultsmentioning

confidence: 99%

“…Now it is known that this metric represents the distance of a geodesic [82] between M and N in the undirected Hasse graph representing the lattice of subspaces of V (2n, 2) partially oredered by inclusion. This notion means that M N if and only if M is a subspace of N .…”

Section: Comments and Speculationsmentioning

confidence: 99%

“…For arbitrary subspaces on the boundary side this information is to be combined with the one provided by the (12) metric d(M, N ). Now it is known that this metric represents the distance of a geodesic [82] between M and N in the undirected Hasse graph representing the lattice of subspaces of V (2n, 2) partially oredered by inclusion. This notion means that M N if and only if M is a subspace of N .…”

Section: Comments and Speculationsmentioning

confidence: 99%

See 1 more Smart Citation

Finite geometric toy model of spacetime as an error correcting code

Lévay

Holweck

2019

Phys. Rev. D

View full text Add to dashboard Cite

A finite geometric model of space-time (which we call the bulk) is shown to emerge as a set of error correcting codes. The bulk is encoding a set of messages located in a blow up of the Gibbons-Hoffman-Wootters (GHW) discrete phase space for n-qubits (which we call the boundary). Our error correcting code is a geometric subspace code known from network coding, and the correspondence map is the finite geometric analogue of the Plücker map well-known form twistor theory. The n = 2 case of the bulk-boundary correspondence is precisely the twistor correspondence where the boundary is playing the role of the twistor space and the bulk is a finite geometric version of compactified Minkowski space-time. For n ≥ 3 the bulk is identified with the finite geometric version of the Brody-Hughston quantum space-time. For special regions on both sides of the correspondence we associate certain collections of qubit observables. On the boundary side this association gives rise to the well-known GHW quantum net structure. In this picture the messages are complete sets of commuting observables associated to Lagrangian subspaces giving a partition of the boundary. Incomplete subsets of observables corresponding to subspaces of the Lagrangian ones are regarded as corrupted messages. Such a partition of the boundary is represented on the bulk side as a special collection of space-time points. For a particular message residing in the boundary, the set of possible errors is described by the fine details of the light-cone structure of its representative space-time point in the bulk. The geometric arrangement of representative space-time points, playing the role of the variety of codewords, encapsulates an algebraic algorithm for recovery from errors on the boundary side.

show abstract

Two characterizations of generalized hypercube

Mollard¹

1991

Discrete Mathematics

View full text Add to dashboard Cite

A q-analogous of the characterization of hypercubes as graphs

Cited by 11 publications

References 2 publications

Coding for Errors and Erasures in Random Network Coding

Coding for Errors and Erasures in Random Network Coding

Finite geometric toy model of spacetime as an error correcting code

Two characterizations of generalized hypercube

Contact Info

Product

Resources

About