Best Known (41−19, 41, s)-Nets in Base 25
(41−19, 41, 178)-Net over F25 — Constructive and digital
Digital (22, 41, 178)-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 (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 (3, 12, 52)-net over F25, using
(41−19, 41, 574)-Net over F25 — Digital
Digital (22, 41, 574)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2541, 574, F25, 19) (dual of [574, 533, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(2541, 639, F25, 19) (dual of [639, 598, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(13) [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(2527, 625, F25, 14) (dual of [625, 598, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(254, 14, F25, 4) (dual of [14, 10, 5]-code or 14-arc in PG(3,25)), using
- discarding factors / shortening the dual code based on linear OA(254, 25, F25, 4) (dual of [25, 21, 5]-code or 25-arc in PG(3,25)), using
- Reed–Solomon code RS(21,25) [i]
- discarding factors / shortening the dual code based on linear OA(254, 25, F25, 4) (dual of [25, 21, 5]-code or 25-arc in PG(3,25)), using
- construction X applied to Ce(18) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(2541, 639, F25, 19) (dual of [639, 598, 20]-code), using
(41−19, 41, 282228)-Net in Base 25 — Upper bound on s
There is no (22, 41, 282229)-net in base 25, because
- 1 times m-reduction [i] would yield (22, 40, 282229)-net in base 25, but
- the generalized Rao bound for nets shows that 25m ≥ 82 720559 801749 404104 759027 450492 112028 661731 613416 260025 > 2540 [i]