Best Known (64, 64+25, s)-Nets in Base 64
(64, 64+25, 21949)-Net over F64 — Constructive and digital
Digital (64, 89, 21949)-net over F64, using
- 641 times duplication [i] based on digital (63, 88, 21949)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (3, 15, 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 (48, 73, 21845)-net over F64, using
- net defined by OOA [i] based on linear OOA(6473, 21845, F64, 25, 25) (dual of [(21845, 25), 546052, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(6473, 262141, F64, 25) (dual of [262141, 262068, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(6473, 262144, F64, 25) (dual of [262144, 262071, 26]-code), using
- an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- discarding factors / shortening the dual code based on linear OA(6473, 262144, F64, 25) (dual of [262144, 262071, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(6473, 262141, F64, 25) (dual of [262141, 262068, 26]-code), using
- net defined by OOA [i] based on linear OOA(6473, 21845, F64, 25, 25) (dual of [(21845, 25), 546052, 26]-NRT-code), using
- digital (3, 15, 104)-net over F64, using
- (u, u+v)-construction [i] based on
(64, 64+25, 174763)-Net in Base 64 — Constructive
(64, 89, 174763)-net in base 64, using
- 642 times duplication [i] based on (62, 87, 174763)-net in base 64, using
- net defined by OOA [i] based on OOA(6487, 174763, S64, 25, 25), using
- OOA 12-folding and stacking with additional row [i] based on OA(6487, 2097157, S64, 25), using
- discarding factors based on OA(6487, 2097160, S64, 25), using
- discarding parts of the base [i] based on linear OA(12874, 2097160, F128, 25) (dual of [2097160, 2097086, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,11]) [i] based on
- linear OA(12873, 2097153, F128, 25) (dual of [2097153, 2097080, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(12867, 2097153, F128, 23) (dual of [2097153, 2097086, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- linear OA(1281, 7, F128, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(1281, s, F128, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,12]) ⊂ C([0,11]) [i] based on
- discarding parts of the base [i] based on linear OA(12874, 2097160, F128, 25) (dual of [2097160, 2097086, 26]-code), using
- discarding factors based on OA(6487, 2097160, S64, 25), using
- OOA 12-folding and stacking with additional row [i] based on OA(6487, 2097157, S64, 25), using
- net defined by OOA [i] based on OOA(6487, 174763, S64, 25, 25), using
(64, 64+25, 776151)-Net over F64 — Digital
Digital (64, 89, 776151)-net over F64, using
(64, 64+25, large)-Net in Base 64 — Upper bound on s
There is no (64, 89, large)-net in base 64, because
- 23 times m-reduction [i] would yield (64, 66, large)-net in base 64, but