Best Known (67, 67+24, s)-Nets in Base 64
(67, 67+24, 22023)-Net over F64 — Constructive and digital
Digital (67, 91, 22023)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (7, 19, 177)-net over F64, using
- net from sequence [i] based on digital (7, 176)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 7 and N(F) ≥ 177, using
- net from sequence [i] based on digital (7, 176)-sequence over F64, using
- digital (48, 72, 21846)-net over F64, using
- net defined by OOA [i] based on linear OOA(6472, 21846, F64, 24, 24) (dual of [(21846, 24), 524232, 25]-NRT-code), using
- OA 12-folding and stacking [i] based on linear OA(6472, 262152, F64, 24) (dual of [262152, 262080, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(6472, 262155, F64, 24) (dual of [262155, 262083, 25]-code), using
- construction X applied to Ce(23) ⊂ Ce(20) [i] based on
- linear OA(6470, 262144, F64, 24) (dual of [262144, 262074, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(6461, 262144, F64, 21) (dual of [262144, 262083, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(642, 11, F64, 2) (dual of [11, 9, 3]-code or 11-arc in PG(1,64)), using
- discarding factors / shortening the dual code based on linear OA(642, 64, F64, 2) (dual of [64, 62, 3]-code or 64-arc in PG(1,64)), using
- Reed–Solomon code RS(62,64) [i]
- discarding factors / shortening the dual code based on linear OA(642, 64, F64, 2) (dual of [64, 62, 3]-code or 64-arc in PG(1,64)), using
- construction X applied to Ce(23) ⊂ Ce(20) [i] based on
- discarding factors / shortening the dual code based on linear OA(6472, 262155, F64, 24) (dual of [262155, 262083, 25]-code), using
- OA 12-folding and stacking [i] based on linear OA(6472, 262152, F64, 24) (dual of [262152, 262080, 25]-code), using
- net defined by OOA [i] based on linear OOA(6472, 21846, F64, 24, 24) (dual of [(21846, 24), 524232, 25]-NRT-code), using
- digital (7, 19, 177)-net over F64, using
(67, 67+24, 174765)-Net in Base 64 — Constructive
(67, 91, 174765)-net in base 64, using
- base change [i] based on digital (54, 78, 174765)-net over F128, using
- 1281 times duplication [i] based on digital (53, 77, 174765)-net over F128, using
- net defined by OOA [i] based on linear OOA(12877, 174765, F128, 24, 24) (dual of [(174765, 24), 4194283, 25]-NRT-code), using
- OA 12-folding and stacking [i] based on linear OA(12877, 2097180, F128, 24) (dual of [2097180, 2097103, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(12877, 2097183, F128, 24) (dual of [2097183, 2097106, 25]-code), using
- construction X applied to Ce(23) ⊂ Ce(15) [i] based on
- linear OA(12870, 2097152, F128, 24) (dual of [2097152, 2097082, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(12846, 2097152, F128, 16) (dual of [2097152, 2097106, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(1287, 31, F128, 7) (dual of [31, 24, 8]-code or 31-arc in PG(6,128)), using
- discarding factors / shortening the dual code based on linear OA(1287, 128, F128, 7) (dual of [128, 121, 8]-code or 128-arc in PG(6,128)), using
- Reed–Solomon code RS(121,128) [i]
- discarding factors / shortening the dual code based on linear OA(1287, 128, F128, 7) (dual of [128, 121, 8]-code or 128-arc in PG(6,128)), using
- construction X applied to Ce(23) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(12877, 2097183, F128, 24) (dual of [2097183, 2097106, 25]-code), using
- OA 12-folding and stacking [i] based on linear OA(12877, 2097180, F128, 24) (dual of [2097180, 2097103, 25]-code), using
- net defined by OOA [i] based on linear OOA(12877, 174765, F128, 24, 24) (dual of [(174765, 24), 4194283, 25]-NRT-code), using
- 1281 times duplication [i] based on digital (53, 77, 174765)-net over F128, using
(67, 67+24, 2095600)-Net over F64 — Digital
Digital (67, 91, 2095600)-net over F64, using
(67, 67+24, large)-Net in Base 64 — Upper bound on s
There is no (67, 91, large)-net in base 64, because
- 22 times m-reduction [i] would yield (67, 69, large)-net in base 64, but