Arcs and Ovals in the Hermitian and Ree Unitals

The hermitian unitals U(q) and the Ree unitals RU(q) are examined for the existence of ovals and arcs. It is shown that U(q) does not have ovals for q > 2 and that RU(q), like U(q), is embedded in a much larger design with block intersections of cardinality ⩽ 2. Arcs of size 3q + 1 are constructed for the Ree unitals RU(q); they are ovals only in the case q = 3. In this case, U(3) and RU(3) are embedded in the same design and its automorphism group, the symplectic group Sp(6, 2), contains the automorphism groups of both the unitals; the coding-theoretic aspects are elucidated

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

On the Reed-Muller codes

AbstractWe give a brief but complete account of all the essential facts concerning the Reed-Muller and punctured Reed-Muller codes. The treatment is new and includes an easy, direct proof of the fact that the punctured Reed-Muller codes are the codes of the projective geometries over the binary field. We also establish the existence of two short exact sequences that lead to new proofs that the minimum-weight vectors of the Reed-Muller and punctured Reed–Muller codes are the incidence vectors of the appropriate geometric objects