Error-correcting codes on projective bundles

Algebraic Geometry Codes from Higher Dimensional Varieties

“…Codes from ruled surfaces were also considered in Lomont's thesis, [39]. The results for codes over ruled surfaces have been generalized to give corresponding results for codes on projective bundles P(E) for E of all ranks r ≥ 2 by Nakashima in [42]. Nakashima also considers codes on Grassmann, quadric, and Hermitian bundles in [41].…”

Scroll codes over curves of higher genus

“…The following proposition is the RiemannRoch theorem for vector bundles on curves dened over nite elds, and is used repeatedly by other authors, like in [Han01] and [Nak06].…”

“…(a) In the case g = 1, j = 2, q = 4, we have an elliptic curve, and according to an unpublished PhD thesis by Agnes Tillmann, whose proof is recalled in [AEB92] (see also the proof of Corollary 3.1 of [Nak06]), there exists a canonical semistable vector bundle E d,r dened over F q of degree d and rank r for all integers d ∈ Z and r ≥ 1, and hence in particular for all integers d and r that we are interested in.…”

“…In other papers, like [Han01], [Lom03] and [Nak06], one also studies projective bundles T = P(E ) like this for the purpose of producing codes, and one even varies the complete linear system line bundle aL + f 1 + · · · + f b by which one embeds P(E ) into projective space, where L is as described, and the f j are bres of P(E ) over points P 1 , · · · , P b on X. There one gives estimates for the minimum distance d 1 for the codes thus dened, in other words for (the number of points minus) the maximal number of F q -rational points in a codimension one space in the embedding space.…”

Scroll codes

“…They give lower bounds for the minimum distance of the codes produced, and do not treat the higher weights. In [3], one studies projectivizations of semistable bundles of arbitrary rank over curves of genus at least two. Nakashima obtains good estimates for the minimum distance of the codes produced.…”