Best Known (39, 39+12, s)-Nets in Base 64
(39, 39+12, 1398165)-Net over F64 — Constructive and digital
Digital (39, 51, 1398165)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (0, 6, 65)-net over F64, using
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 0 and N(F) ≥ 65, using
- the rational function field F64(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
- digital (33, 45, 1398100)-net over F64, using
- net defined by OOA [i] based on linear OOA(6445, 1398100, F64, 12, 12) (dual of [(1398100, 12), 16777155, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(6445, 8388600, F64, 12) (dual of [8388600, 8388555, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(6445, large, F64, 12) (dual of [large, large−45, 13]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 644−1, defining interval I = [0,11], and designed minimum distance d ≥ |I|+1 = 13 [i]
- discarding factors / shortening the dual code based on linear OA(6445, large, F64, 12) (dual of [large, large−45, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(6445, 8388600, F64, 12) (dual of [8388600, 8388555, 13]-code), using
- net defined by OOA [i] based on linear OOA(6445, 1398100, F64, 12, 12) (dual of [(1398100, 12), 16777155, 13]-NRT-code), using
- digital (0, 6, 65)-net over F64, using
(39, 39+12, large)-Net over F64 — Digital
Digital (39, 51, large)-net over F64, using
- 641 times duplication [i] based on digital (38, 50, large)-net over F64, using
- t-expansion [i] based on digital (37, 50, large)-net over F64, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(6450, large, F64, 13) (dual of [large, large−50, 14]-code), using
- 1 times code embedding in larger space [i] based on linear OA(6449, large, F64, 13) (dual of [large, large−49, 14]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 648−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- 1 times code embedding in larger space [i] based on linear OA(6449, large, F64, 13) (dual of [large, large−49, 14]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(6450, large, F64, 13) (dual of [large, large−50, 14]-code), using
- t-expansion [i] based on digital (37, 50, large)-net over F64, using
(39, 39+12, large)-Net in Base 64 — Upper bound on s
There is no (39, 51, large)-net in base 64, because
- 10 times m-reduction [i] would yield (39, 41, large)-net in base 64, but