Best Known (68, 68+16, s)-Nets in Base 25
(68, 68+16, 1048601)-Net over F25 — Constructive and digital
Digital (68, 84, 1048601)-net over F25, using
- (u, u+v)-construction [i] based on
- digital (0, 8, 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 (60, 76, 1048575)-net over F25, using
- net defined by OOA [i] based on linear OOA(2576, 1048575, F25, 16, 16) (dual of [(1048575, 16), 16777124, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(2576, 8388600, F25, 16) (dual of [8388600, 8388524, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(2576, large, F25, 16) (dual of [large, large−76, 17]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 9765624 = 255−1, defining interval I = [0,15], and designed minimum distance d ≥ |I|+1 = 17 [i]
- discarding factors / shortening the dual code based on linear OA(2576, large, F25, 16) (dual of [large, large−76, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(2576, 8388600, F25, 16) (dual of [8388600, 8388524, 17]-code), using
- net defined by OOA [i] based on linear OOA(2576, 1048575, F25, 16, 16) (dual of [(1048575, 16), 16777124, 17]-NRT-code), using
- digital (0, 8, 26)-net over F25, using
(68, 68+16, large)-Net over F25 — Digital
Digital (68, 84, large)-net over F25, using
- 252 times duplication [i] based on digital (66, 82, large)-net over F25, using
- t-expansion [i] based on digital (65, 82, large)-net over F25, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(2582, large, F25, 17) (dual of [large, large−82, 18]-code), using
- 1 times code embedding in larger space [i] based on linear OA(2581, large, F25, 17) (dual of [large, large−81, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 9765626 | 2510−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- 1 times code embedding in larger space [i] based on linear OA(2581, large, F25, 17) (dual of [large, large−81, 18]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(2582, large, F25, 17) (dual of [large, large−82, 18]-code), using
- t-expansion [i] based on digital (65, 82, large)-net over F25, using
(68, 68+16, large)-Net in Base 25 — Upper bound on s
There is no (68, 84, large)-net in base 25, because
- 14 times m-reduction [i] would yield (68, 70, large)-net in base 25, but