Best Known (59, 59+16, s)-Nets in Base 64
(59, 59+16, 1048770)-Net over F64 — Constructive and digital
Digital (59, 75, 1048770)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (6, 14, 195)-net over F64, using
- generalized (u, u+v)-construction [i] based on
- digital (0, 2, 65)-net over F64, using
- digital (0, 4, 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 (0, 8, 65)-net over F64, using
- net from sequence [i] based on digital (0, 64)-sequence over F64 (see above)
- generalized (u, u+v)-construction [i] based on
- digital (45, 61, 1048575)-net over F64, using
- net defined by OOA [i] based on linear OOA(6461, 1048575, F64, 16, 16) (dual of [(1048575, 16), 16777139, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(6461, 8388600, F64, 16) (dual of [8388600, 8388539, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(6461, large, F64, 16) (dual of [large, large−61, 17]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 644−1, defining interval I = [0,15], and designed minimum distance d ≥ |I|+1 = 17 [i]
- discarding factors / shortening the dual code based on linear OA(6461, large, F64, 16) (dual of [large, large−61, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(6461, 8388600, F64, 16) (dual of [8388600, 8388539, 17]-code), using
- net defined by OOA [i] based on linear OOA(6461, 1048575, F64, 16, 16) (dual of [(1048575, 16), 16777139, 17]-NRT-code), using
- digital (6, 14, 195)-net over F64, using
(59, 59+16, 1048834)-Net in Base 64 — Constructive
(59, 75, 1048834)-net in base 64, using
- (u, u+v)-construction [i] based on
- (6, 14, 259)-net in base 64, using
- 2 times m-reduction [i] based on (6, 16, 259)-net in base 64, using
- base change [i] based on digital (2, 12, 259)-net over F256, using
- net from sequence [i] based on digital (2, 258)-sequence over F256, using
- base change [i] based on digital (2, 12, 259)-net over F256, using
- 2 times m-reduction [i] based on (6, 16, 259)-net in base 64, using
- digital (45, 61, 1048575)-net over F64, using
- net defined by OOA [i] based on linear OOA(6461, 1048575, F64, 16, 16) (dual of [(1048575, 16), 16777139, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(6461, 8388600, F64, 16) (dual of [8388600, 8388539, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(6461, large, F64, 16) (dual of [large, large−61, 17]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 644−1, defining interval I = [0,15], and designed minimum distance d ≥ |I|+1 = 17 [i]
- discarding factors / shortening the dual code based on linear OA(6461, large, F64, 16) (dual of [large, large−61, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(6461, 8388600, F64, 16) (dual of [8388600, 8388539, 17]-code), using
- net defined by OOA [i] based on linear OOA(6461, 1048575, F64, 16, 16) (dual of [(1048575, 16), 16777139, 17]-NRT-code), using
- (6, 14, 259)-net in base 64, using
(59, 59+16, large)-Net over F64 — Digital
Digital (59, 75, large)-net over F64, using
- t-expansion [i] based on digital (57, 75, large)-net over F64, using
- 1 times m-reduction [i] based on digital (57, 76, large)-net over F64, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(6476, large, F64, 19) (dual of [large, large−76, 20]-code), using
- 3 times code embedding in larger space [i] 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]
- 3 times code embedding in larger space [i] based on linear OA(6473, large, F64, 19) (dual of [large, large−73, 20]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(6476, large, F64, 19) (dual of [large, large−76, 20]-code), using
- 1 times m-reduction [i] based on digital (57, 76, large)-net over F64, using
(59, 59+16, large)-Net in Base 64 — Upper bound on s
There is no (59, 75, large)-net in base 64, because
- 14 times m-reduction [i] would yield (59, 61, large)-net in base 64, but