Best Known (67−27, 67, s)-Nets in Base 64
(67−27, 67, 513)-Net over F64 — Constructive and digital
Digital (40, 67, 513)-net over F64, using
- t-expansion [i] based on digital (28, 67, 513)-net over F64, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- the Hermitian function field over F64 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
(67−27, 67, 1261)-Net in Base 64 — Constructive
(40, 67, 1261)-net in base 64, using
- 641 times duplication [i] based on (39, 66, 1261)-net in base 64, using
- net defined by OOA [i] based on OOA(6466, 1261, S64, 27, 27), using
- OOA 13-folding and stacking with additional row [i] based on OA(6466, 16394, S64, 27), using
- discarding factors based on OA(6466, 16396, S64, 27), using
- discarding parts of the base [i] based on linear OA(12856, 16396, F128, 27) (dual of [16396, 16340, 28]-code), using
- construction X applied to C([0,13]) ⊂ C([0,11]) [i] based on
- linear OA(12853, 16385, F128, 27) (dual of [16385, 16332, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(12845, 16385, F128, 23) (dual of [16385, 16340, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- linear OA(1283, 11, F128, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,128) or 11-cap in PG(2,128)), using
- discarding factors / shortening the dual code based on linear OA(1283, 128, F128, 3) (dual of [128, 125, 4]-code or 128-arc in PG(2,128) or 128-cap in PG(2,128)), using
- Reed–Solomon code RS(125,128) [i]
- discarding factors / shortening the dual code based on linear OA(1283, 128, F128, 3) (dual of [128, 125, 4]-code or 128-arc in PG(2,128) or 128-cap in PG(2,128)), using
- construction X applied to C([0,13]) ⊂ C([0,11]) [i] based on
- discarding parts of the base [i] based on linear OA(12856, 16396, F128, 27) (dual of [16396, 16340, 28]-code), using
- discarding factors based on OA(6466, 16396, S64, 27), using
- OOA 13-folding and stacking with additional row [i] based on OA(6466, 16394, S64, 27), using
- net defined by OOA [i] based on OOA(6466, 1261, S64, 27, 27), using
(67−27, 67, 7570)-Net over F64 — Digital
Digital (40, 67, 7570)-net over F64, using
(67−27, 67, large)-Net in Base 64 — Upper bound on s
There is no (40, 67, large)-net in base 64, because
- 25 times m-reduction [i] would yield (40, 42, large)-net in base 64, but