64 research outputs found
m-systems of polar spaces and SPG reguli
It will be shown that every m-system of W2n+1(q), Q(-)(2n+1,q) or H(2n, q(2)) is an SPG regulus and hence gives rise to a semipartial geometry. We also briefly investigate the semipartial geometries, associated with the known m-systems of these polar spaces
LDPC codes from semipartial geometries
A binary low-density parity-check (LDPC) code is a linear block code that is defined by a sparse parity-check matrix H, that is H has a low density of 1’s. LDPC codes were originally presented by Gallager in his doctoral dissertation [9], but largely overlooked for the next 35 years. A notable exception was [29], in which Tanner introduced a graphical representation for LDPC codes, now known as Tanner graphs. However, interest in these codes has greatly increased since 1996 with the publication of [22] and other papers, since it has been realised that LDPC codes are capable of achieving near-optimal performance when decoded using iterative decoding algorithms.
LDPC codes can be constructed randomly by using a computer algorithm to generate a suitable matrix H. However, it is also possible to construct LDPC codes explicitly using various incidence structures in discrete mathematics. For example, LDPC codes can be constructed based on the points and lines of finite geometries: there are many examples in the literature (see for example [18, 28]). These constructed codes can possess certain advantages over randomly-generated codes. For example they may provide more efficient encoding algorithms than randomly-generated codes. Furthermore it can be easier to understand and determine the properties of such codes because of the underlying structure.
LDPC codes have been constructed based on incidence structures known as partial geometries [16]. The aim of this research is to provide examples of new codes constructed based on structures known as semipartial geometries (SPGs), which are generalisations of partial geometries.
Since the commencement of this thesis [19] was published, which showed that codes could be constructed from semipartial geometries and provided some examples and basic results. By necessity this thesis contains a number of results from that paper. However, it should be noted that the scope of [19] is fairly limited and that the overlap between the current thesis and [19] is consequently small. [19] also contains a number of errors, some of which have been noted and corrected in this thesis
Perp-systems and partial geometries
A perp-system R(r) is a maximal set of r-dimensional subspaces of PG(N,q) equipped with a polarity rho, such that the tangent space of an element of R(r) does not intersect any element of R(r). We prove that a perp-system yields partial geometries, strongly regular graphs, two-weight codes, maximal arcs and k-ovoids. We also give some examples, one of them yielding a new pg(8,20,2)
New Classes of Partial Geometries and Their Associated LDPC Codes
The use of partial geometries to construct parity-check matrices for LDPC
codes has resulted in the design of successful codes with a probability of
error close to the Shannon capacity at bit error rates down to . Such
considerations have motivated this further investigation. A new and simple
construction of a type of partial geometries with quasi-cyclic structure is
given and their properties are investigated. The trapping sets of the partial
geometry codes were considered previously using the geometric aspects of the
underlying structure to derive information on the size of allowable trapping
sets. This topic is further considered here. Finally, there is a natural
relationship between partial geometries and strongly regular graphs. The
eigenvalues of the adjacency matrices of such graphs are well known and it is
of interest to determine if any of the Tanner graphs derived from the partial
geometries are good expanders for certain parameter sets, since it can be
argued that codes with good geometric and expansion properties might perform
well under message-passing decoding.Comment: 34 pages with single column, 6 figure
The isomorphism problem for linear representations and their graphs
In this paper, we study the isomorphism problem for linear representations. A
linear representation Tn*(K) of a point set K is a point-line geometry,
embedded in a projective space PG(n+1,q), where K is contained in a hyperplane.
We put constraints on K which ensure that every automorphism of Tn*(K) is
induced by a collineation of the ambient projective space. This allows us to
show that, under certain conditions, two linear representations Tn*(K) and
Tn*(K') are isomorphic if and only if the point sets K and K' are
PGammaL-equivalent. We also deal with the slightly more general problem of
isomorphic incidence graphs of linear representations. In the last part of this
paper, we give an explicit description of the group of automorphisms of Tn*(K)
that are induced by collineations of PG(n+1,q).Comment: 14 page
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