Best Known (21, 21+19, s)-Nets in Base 25
(21, 21+19, 156)-Net over F25 — Constructive and digital
Digital (21, 40, 156)-net over F25, using
- (u, u+v)-construction [i] based on
- digital (3, 12, 52)-net over F25, using
- net from sequence [i] based on digital (3, 51)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 3 and N(F) ≥ 52, using
- net from sequence [i] based on digital (3, 51)-sequence over F25, using
- digital (9, 28, 104)-net over F25, using
- net from sequence [i] based on digital (9, 103)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 9 and N(F) ≥ 104, using
- net from sequence [i] based on digital (9, 103)-sequence over F25, using
- digital (3, 12, 52)-net over F25, using
(21, 21+19, 474)-Net over F25 — Digital
Digital (21, 40, 474)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2540, 474, F25, 19) (dual of [474, 434, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(2540, 637, F25, 19) (dual of [637, 597, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,7]) [i] based on
- linear OA(2537, 626, F25, 19) (dual of [626, 589, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 626 | 254−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(2529, 626, F25, 15) (dual of [626, 597, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 626 | 254−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (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([0,9]) ⊂ C([0,7]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2540, 637, F25, 19) (dual of [637, 597, 20]-code), using
(21, 21+19, 197365)-Net in Base 25 — Upper bound on s
There is no (21, 40, 197366)-net in base 25, because
- 1 times m-reduction [i] would yield (21, 39, 197366)-net in base 25, but
- the generalized Rao bound for nets shows that 25m ≥ 3 308840 542739 478478 985662 382244 281526 125121 133479 705745 > 2539 [i]