Intersection sets, three-character multisets and associated codes

Abstract

In this article we construct new minimal intersection sets in AG(r, q 2 ) sporting three intersection numbers with hyperplanes; we then use these sets to obtain linear error correcting codes with few weights, whose weight enumerator we also determine. Furthermore, we provide a new family of three-character multisets in PG(r, q 2 ) with r even and we also compute their weight distribution.

Codes with few weights arising from linear sets

“…Our construction provides codes for which this property does not hold. When r > 3 the codes we construct are q-divisible, whereas for r = 3 they are not, contrary to those presented in [1]. Furthermore, the codes introduced in [1] are defined over F q 2 , whereas our construction yield codes defined over F q n , for any n ≥ 3.…”

“…those arising from (multi-)set with three intersection numbers with respect to hyperplanes. A general geometric construction for three-weight codes has been given in [1], by using the property of the Hermitian variety. The codes they obtained have the property that all the nonzero weights are divisible by q.…”

“…, t}. If G is a (3 × N )-matrix G over F q n whose columns correspond to the following set of vectors {(f 1 1 (x 1 ), f 1 2 (x 1 ), f 1 3 (x 1 ), . .…”

Codes with few weights arising from linear sets

“…[13,14,15,33,35], there are few known geometric constructions [19]. A general geometric construction for three-weight codes has been given in [1]. Let a ∈ F * q 2 , b ∈ F q 2 \F q and B(a, b) be the affine algebraic set of AG(r −1, q 2 ) of equation…”

“…For the family of three-weight codes introduced above, we have that 3 divides rn and their length does not divide the order of the field minus one, which make these codes not equivalent to most of the known three-weight codes, see [13,Section VI]. As for the codes presented in [1], when r > 3 the codes we construct are q-divisible, whereas for r = 3 this does not hold and hence in such a case the codes are not equivalent to those in [1]. Since the construction of [1] works for n even and r odd and since for r > 3 it gives codes with length divisible by q, our construction provides a large family of three-weight codes not equivalent to the aforementioned ones.…”