Best Known (68, 68+14, s)-Nets in Base 81
(68, 68+14, 2396824)-Net over F81 — Constructive and digital
Digital (68, 82, 2396824)-net over F81, using
- generalized (u, u+v)-construction [i] based on
- digital (0, 4, 82)-net over F81, using
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 0 and N(F) ≥ 82, using
- the rational function field F81(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- digital (18, 25, 1198371)-net over F81, using
- s-reduction based on digital (18, 25, 2796200)-net over F81, using
- net defined by OOA [i] based on linear OOA(8125, 2796200, F81, 7, 7) (dual of [(2796200, 7), 19573375, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(8125, 8388601, F81, 7) (dual of [8388601, 8388576, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(8125, large, F81, 7) (dual of [large, large−25, 8]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523361 | 818−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(8125, large, F81, 7) (dual of [large, large−25, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(8125, 8388601, F81, 7) (dual of [8388601, 8388576, 8]-code), using
- net defined by OOA [i] based on linear OOA(8125, 2796200, F81, 7, 7) (dual of [(2796200, 7), 19573375, 8]-NRT-code), using
- s-reduction based on digital (18, 25, 2796200)-net over F81, using
- digital (39, 53, 1198371)-net over F81, using
- net defined by OOA [i] based on linear OOA(8153, 1198371, F81, 14, 14) (dual of [(1198371, 14), 16777141, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(8153, 8388597, F81, 14) (dual of [8388597, 8388544, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(8153, large, F81, 14) (dual of [large, large−53, 15]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523360 | 814−1, defining interval I = [0,13], and designed minimum distance d ≥ |I|+1 = 15 [i]
- discarding factors / shortening the dual code based on linear OA(8153, large, F81, 14) (dual of [large, large−53, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(8153, 8388597, F81, 14) (dual of [8388597, 8388544, 15]-code), using
- net defined by OOA [i] based on linear OOA(8153, 1198371, F81, 14, 14) (dual of [(1198371, 14), 16777141, 15]-NRT-code), using
- digital (0, 4, 82)-net over F81, using
(68, 68+14, large)-Net over F81 — Digital
Digital (68, 82, large)-net over F81, using
- 811 times duplication [i] based on digital (67, 81, large)-net over F81, using
- t-expansion [i] based on digital (60, 81, large)-net over F81, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8181, large, F81, 21) (dual of [large, large−81, 22]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523361 | 818−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8181, large, F81, 21) (dual of [large, large−81, 22]-code), using
- t-expansion [i] based on digital (60, 81, large)-net over F81, using
(68, 68+14, large)-Net in Base 81 — Upper bound on s
There is no (68, 82, large)-net in base 81, because
- 12 times m-reduction [i] would yield (68, 70, large)-net in base 81, but