Best Known (68, 68+12, s)-Nets in Base 25
(68, 68+12, 1528335)-Net over F25 — Constructive and digital
Digital (68, 80, 1528335)-net over F25, using
- (u, u+v)-construction [i] based on
- digital (18, 24, 130235)-net over F25, using
- (u, u+v)-construction [i] based on
- digital (0, 3, 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 (15, 21, 130209)-net over F25, using
- net defined by OOA [i] based on linear OOA(2521, 130209, F25, 6, 6) (dual of [(130209, 6), 781233, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(2521, 390627, F25, 6) (dual of [390627, 390606, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(2521, 390629, F25, 6) (dual of [390629, 390608, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- linear OA(2521, 390625, F25, 6) (dual of [390625, 390604, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 390624 = 254−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(2517, 390625, F25, 5) (dual of [390625, 390608, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 390624 = 254−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(250, 4, F25, 0) (dual of [4, 4, 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 Ce(5) ⊂ Ce(4) [i] based on
- discarding factors / shortening the dual code based on linear OA(2521, 390629, F25, 6) (dual of [390629, 390608, 7]-code), using
- OA 3-folding and stacking [i] based on linear OA(2521, 390627, F25, 6) (dual of [390627, 390606, 7]-code), using
- net defined by OOA [i] based on linear OOA(2521, 130209, F25, 6, 6) (dual of [(130209, 6), 781233, 7]-NRT-code), using
- digital (0, 3, 26)-net over F25, using
- (u, u+v)-construction [i] based on
- digital (44, 56, 1398100)-net over F25, using
- net defined by OOA [i] based on linear OOA(2556, 1398100, F25, 12, 12) (dual of [(1398100, 12), 16777144, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(2556, 8388600, F25, 12) (dual of [8388600, 8388544, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(2556, large, F25, 12) (dual of [large, large−56, 13]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 9765624 = 255−1, defining interval I = [0,11], and designed minimum distance d ≥ |I|+1 = 13 [i]
- discarding factors / shortening the dual code based on linear OA(2556, large, F25, 12) (dual of [large, large−56, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(2556, 8388600, F25, 12) (dual of [8388600, 8388544, 13]-code), using
- net defined by OOA [i] based on linear OOA(2556, 1398100, F25, 12, 12) (dual of [(1398100, 12), 16777144, 13]-NRT-code), using
- digital (18, 24, 130235)-net over F25, using
(68, 68+12, large)-Net over F25 — Digital
Digital (68, 80, large)-net over F25, using
- t-expansion [i] based on digital (65, 80, large)-net over F25, using
- 2 times m-reduction [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
- 2 times m-reduction [i] based on digital (65, 82, large)-net over F25, using
(68, 68+12, large)-Net in Base 25 — Upper bound on s
There is no (68, 80, large)-net in base 25, because
- 10 times m-reduction [i] would yield (68, 70, large)-net in base 25, but