Best Known (15, 15+13, s)-Nets in Base 25
(15, 15+13, 132)-Net over F25 — Constructive and digital
Digital (15, 28, 132)-net over F25, using
- 1 times m-reduction [i] based on digital (15, 29, 132)-net over F25, using
- (u, u+v)-construction [i] based on
- digital (4, 11, 66)-net over F25, using
- net from sequence [i] based on digital (4, 65)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 4 and N(F) ≥ 66, using
- net from sequence [i] based on digital (4, 65)-sequence over F25, using
- digital (4, 18, 66)-net over F25, using
- net from sequence [i] based on digital (4, 65)-sequence over F25 (see above)
- digital (4, 11, 66)-net over F25, using
- (u, u+v)-construction [i] based on
(15, 15+13, 547)-Net over F25 — Digital
Digital (15, 28, 547)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2528, 547, F25, 13) (dual of [547, 519, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(2528, 637, F25, 13) (dual of [637, 609, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([0,4]) [i] based on
- linear OA(2525, 626, F25, 13) (dual of [626, 601, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 626 | 254−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(2517, 626, F25, 9) (dual of [626, 609, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 626 | 254−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(253, 11, F25, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,25) or 11-cap in PG(2,25)), using
- discarding factors / shortening the dual code based on linear OA(253, 25, F25, 3) (dual of [25, 22, 4]-code or 25-arc in PG(2,25) or 25-cap in PG(2,25)), using
- Reed–Solomon code RS(22,25) [i]
- discarding factors / shortening the dual code based on linear OA(253, 25, F25, 3) (dual of [25, 22, 4]-code or 25-arc in PG(2,25) or 25-cap in PG(2,25)), using
- construction X applied to C([0,6]) ⊂ C([0,4]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2528, 637, F25, 13) (dual of [637, 609, 14]-code), using
(15, 15+13, 243632)-Net in Base 25 — Upper bound on s
There is no (15, 28, 243633)-net in base 25, because
- 1 times m-reduction [i] would yield (15, 27, 243633)-net in base 25, but
- the generalized Rao bound for nets shows that 25m ≥ 55 511254 737203 721846 576096 424355 474257 > 2527 [i]