Best Known (39−19, 39, s)-Nets in Base 25
(39−19, 39, 153)-Net over F25 — Constructive and digital
Digital (20, 39, 153)-net over F25, using
- (u, u+v)-construction [i] based on
- digital (1, 10, 27)-net over F25, using
- net from sequence [i] based on digital (1, 26)-sequence over F25, using
- digital (10, 29, 126)-net over F25, using
- net from sequence [i] based on digital (10, 125)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 10 and N(F) ≥ 126, using
- the Hermitian function field over F25 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 10 and N(F) ≥ 126, using
- net from sequence [i] based on digital (10, 125)-sequence over F25, using
- digital (1, 10, 27)-net over F25, using
(39−19, 39, 391)-Net over F25 — Digital
Digital (20, 39, 391)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2539, 391, F25, 19) (dual of [391, 352, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(2539, 633, F25, 19) (dual of [633, 594, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(15) [i] based on
- linear OA(2537, 625, F25, 19) (dual of [625, 588, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(2531, 625, F25, 16) (dual of [625, 594, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(252, 8, F25, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,25)), using
- discarding factors / shortening the dual code based on linear OA(252, 25, F25, 2) (dual of [25, 23, 3]-code or 25-arc in PG(1,25)), using
- Reed–Solomon code RS(23,25) [i]
- discarding factors / shortening the dual code based on linear OA(252, 25, F25, 2) (dual of [25, 23, 3]-code or 25-arc in PG(1,25)), using
- construction X applied to Ce(18) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(2539, 633, F25, 19) (dual of [633, 594, 20]-code), using
(39−19, 39, 138019)-Net in Base 25 — Upper bound on s
There is no (20, 39, 138020)-net in base 25, because
- 1 times m-reduction [i] would yield (20, 38, 138020)-net in base 25, but
- the generalized Rao bound for nets shows that 25m ≥ 132354 967209 748133 950794 969473 361193 517027 378495 692897 > 2538 [i]