Best Known (19−11, 19, s)-Nets in Base 25
(19−11, 19, 78)-Net over F25 — Constructive and digital
Digital (8, 19, 78)-net over F25, using
- (u, u+v)-construction [i] based on
- digital (0, 5, 26)-net over F25, using
- net from sequence [i] based on digital (0, 25)-sequence over F25, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 0 and N(F) ≥ 26, using
- the rational function field F25(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 25)-sequence over F25, using
- digital (3, 14, 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 (0, 5, 26)-net over F25, using
(19−11, 19, 105)-Net over F25 — Digital
Digital (8, 19, 105)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2519, 105, F25, 11) (dual of [105, 86, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(2519, 106, F25, 11) (dual of [106, 87, 12]-code), using
- construction X applied to C([7,17]) ⊂ C([8,17]) [i] based on
- linear OA(2519, 104, F25, 11) (dual of [104, 85, 12]-code), using the BCH-code C(I) with length 104 | 252−1, defining interval I = {7,8,…,17}, and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(2517, 104, F25, 10) (dual of [104, 87, 11]-code), using the BCH-code C(I) with length 104 | 252−1, defining interval I = {8,9,…,17}, and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(250, 2, F25, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(250, s, F25, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to C([7,17]) ⊂ C([8,17]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2519, 106, F25, 11) (dual of [106, 87, 12]-code), using
(19−11, 19, 11698)-Net in Base 25 — Upper bound on s
There is no (8, 19, 11699)-net in base 25, because
- 1 times m-reduction [i] would yield (8, 18, 11699)-net in base 25, but
- the generalized Rao bound for nets shows that 25m ≥ 14 555545 918347 873940 356841 > 2518 [i]