# Read PDF Finite geometry and coding theory

This geometrical structure is important enough to deserve a name. Def in ition 4An n, w; d, q -m in ihyper is a multiset of n po in ts in PG d, q with the property thatevery hyperplane meets it in at least w po in ts. Many characterisation theorems of m in ihypers are known.

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Now assume that H is not a k-space of PG d, q , i. A contradiction, i. H is asubspace.

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We refer to the literaturefor the known results. As a concrete example of a deep characterization result, wemention the follow in g result of Hamada, Helleseth, and Maekawa. Cover in g radiusFor an e-error correct in g code, we search for a large set of pairwise disjo in t spheres of radius e in the Hamm in g space F n q.

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6. The problem of cover in g codes is an oppositeproblem. Here, we wish to cover all the po in ts of the Hamm in g space F n q with as fewspheres as possible. Cover in g codes f in d applications in data compression. Formally we def in e:Def in ition 5Let C be a l in ear [n, k, d] q -code. The cover in g radius of the code C is the smallest in tegerR such that every n-tuple in F n q lies at Hamm in g distance at most R from a codeword in C.

The follow in g theorem will be the basis for mak in g the l in k with the geometricallyequivalent objects of the saturat in g sets in f in ite geometry. Cover in g codes are l in ked to many geometrical objects.

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Obviously the goal of cover in g codewords becomes easier when one can use morecodewords. So we are in terested in small cover in g codes or equivalently in small saturat in gsets. Example 5 Brualdi et al. We give the description viacoord in ates. Figure 5. Thus Q lies on a l in e which meets S in two po in ts. We aga in give the description via coord in ates. Figure 6.

## Finite Geometry And Coding Theory Thesis

Secret shar in gSecret shar in g schemes are the cryptographic equivalents of a vault that needs severalkeys to be opened. In the simplest cases there are n participants and each group of kparticipants can reconstruct the secret, but less than k participants have no way to learnanyth in g about the secret. The value f 0 is thesecret. A set of k participants can reconstruct f by in terpolation. Then they can computethe secret f 0.

Thus they ga in no in formation about f 0. Example for the Shamir secret shar in g schemeMany secret shar in g schemes are constructed by f in ite geometry. For example onecan use arcs to construct a k-out- of -n secret shar in g scheme. Less than k participants see the follow in g: their shares P 1 ,. Thus there is no way to decide which po in t of l isthe secret P 0. Thus we have constructed a k-out- of -n secret shar in g scheme.

One can consider more complex access structures. For example, we want that threestaff members together can open the vault, but also two senior staff members alone canopen the vault. Def in ition 7 formalises the idea of an access structure. Def in ition 7Let P be a set of persons.

Example 8 shows how to realise a k-out- of -n access structure with f in ite geometry. We want to generalise this example. The secret and the shares should be subspaces of a f in ite projective space PG n, q. As in Example 8, the reconstruction of the secret shouldbe done by comput in g the span of the shares.

This leads to the follow in g def in ition. Let e i , the i-th vector of unity, correspond tothe set U i. An unauthorised set of persons U is conta in ed in at least one maximal unauthorisedset U i. If Q is a qualified set of persons then for every maximal unauthorised set U i , Qconta in s a person p i not in U i.

Secret shar in g schemes can also be constructed by error-correct in g codes. Theshare of the participant number i is symbol c i. So any set of k persons can compute the secret c 0.

This is an alternative description of the k-out- of -n secret shar in g scheme from Example8. The use of error-correct in g codes for describ in g secret shar in g schemes motivatesthe follow in g def in ition. 