LDPC codes from Singer cycles

The main goal of coding theory is to devise efficient systems to exploit the full capacity of a communication channel, thus achieving an arbitrarily small error probability. Low Density Parity Check (LDPC) codes are a family of block codes--characterised by admitting a sparse parity check matrix--with good correction capabilities. In the present paper the orbits of subspaces of a finite projective space under the action of a Singer cycle are investigated.Comment: 11 Page

Some results on caps and codes related to orthogonal Grassmannians — a preview

In this note we offer a short summary of some recent results, to be contained in a forthcoming paper [4], on projective caps and linear error correcting codes arising from the Grassmann embedding εgr k of an orthogonal Grassmannian ∆k . More precisely, we consider the codes arising from the projective system determined by εgr k (∆k ) and determine some of their parameters. We also investigate special sets of points of ∆k which are met by any line of ∆k in at most 2 points proving that their image under the Grassmann embedding is a projective cap

Families of twisted tensor product codes

Using geometric properties of the variety \cV_{r,t}, the image under the Grassmannian map of a Desarguesian $(t-1)$-spread of \PG(rt-1,q), we introduce error correcting codes related to the twisted tensor product construction, producing several families of constacyclic codes. We exactly determine the parameters of these codes and characterise the words of minimum weight.Comment: Keywords: Segre Product, Veronesean, Grassmannian, Desarguesian spread, Subgeometry, Twisted Product, Constacyclic error correcting code, Minimum weigh

Codes and caps from orthogonal Grassmannians

In this paper we investigate linear error correcting codes and projective caps related to the Grassmann embedding $\varepsilon_k^{gr}$ of an orthogonal Grassmannian $\Delta_k$. In particular, we determine some of the parameters of the codes arising from the projective system determined by $\varepsilon_k^{gr}(\Delta_k)$. We also study special sets of points of $\Delta_k$ which are met by any line of $\Delta_k$ in at most 2 points and we show that their image under the Grassmann embedding $\varepsilon_k^{gr}$ is a projective cap.Comment: Keywords: Polar Grassmannian; dual polar space; embedding; error correcting code; cap; Hadamard matrix; Sylvester construction (this is a slightly revised version of v2, with updated bibliography