Best Known (66, 66+19, s)-Nets in Base 64
(66, 66+19, 932170)-Net over F64 — Constructive and digital
Digital (66, 85, 932170)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (3, 12, 104)-net over F64, using
- net from sequence [i] based on digital (3, 103)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 3 and N(F) ≥ 104, using
- net from sequence [i] based on digital (3, 103)-sequence over F64, using
- digital (54, 73, 932066)-net over F64, using
- net defined by OOA [i] based on linear OOA(6473, 932066, F64, 19, 19) (dual of [(932066, 19), 17709181, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(6473, 8388595, F64, 19) (dual of [8388595, 8388522, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(6473, large, F64, 19) (dual of [large, large−73, 20]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 648−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(6473, large, F64, 19) (dual of [large, large−73, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(6473, 8388595, F64, 19) (dual of [8388595, 8388522, 20]-code), using
- net defined by OOA [i] based on linear OOA(6473, 932066, F64, 19, 19) (dual of [(932066, 19), 17709181, 20]-NRT-code), using
- digital (3, 12, 104)-net over F64, using
(66, 66+19, 932323)-Net in Base 64 — Constructive
(66, 85, 932323)-net in base 64, using
- (u, u+v)-construction [i] based on
- (3, 12, 257)-net in base 64, using
- base change [i] based on digital (0, 9, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 0 and N(F) ≥ 257, using
- the rational function field F256(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- base change [i] based on digital (0, 9, 257)-net over F256, using
- digital (54, 73, 932066)-net over F64, using
- net defined by OOA [i] based on linear OOA(6473, 932066, F64, 19, 19) (dual of [(932066, 19), 17709181, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(6473, 8388595, F64, 19) (dual of [8388595, 8388522, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(6473, large, F64, 19) (dual of [large, large−73, 20]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 648−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(6473, large, F64, 19) (dual of [large, large−73, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(6473, 8388595, F64, 19) (dual of [8388595, 8388522, 20]-code), using
- net defined by OOA [i] based on linear OOA(6473, 932066, F64, 19, 19) (dual of [(932066, 19), 17709181, 20]-NRT-code), using
- (3, 12, 257)-net in base 64, using
(66, 66+19, large)-Net over F64 — Digital
Digital (66, 85, large)-net over F64, using
- 3 times m-reduction [i] based on digital (66, 88, large)-net over F64, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(6488, large, F64, 22) (dual of [large, large−88, 23]-code), using
- 3 times code embedding in larger space [i] based on linear OA(6485, large, F64, 22) (dual of [large, large−85, 23]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 644−1, defining interval I = [0,21], and designed minimum distance d ≥ |I|+1 = 23 [i]
- 3 times code embedding in larger space [i] based on linear OA(6485, large, F64, 22) (dual of [large, large−85, 23]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(6488, large, F64, 22) (dual of [large, large−88, 23]-code), using
(66, 66+19, large)-Net in Base 64 — Upper bound on s
There is no (66, 85, large)-net in base 64, because
- 17 times m-reduction [i] would yield (66, 68, large)-net in base 64, but