Best Known (44, 44+30, s)-Nets in Base 64
(44, 44+30, 593)-Net over F64 — Constructive and digital
Digital (44, 74, 593)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (1, 16, 80)-net over F64, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 1 and N(F) ≥ 80, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
- digital (28, 58, 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
- digital (1, 16, 80)-net over F64, using
(44, 44+30, 1093)-Net in Base 64 — Constructive
(44, 74, 1093)-net in base 64, using
- 1 times m-reduction [i] based on (44, 75, 1093)-net in base 64, using
- net defined by OOA [i] based on OOA(6475, 1093, S64, 31, 31), using
- OOA 15-folding and stacking with additional row [i] based on OA(6475, 16396, S64, 31), using
- discarding parts of the base [i] based on linear OA(12864, 16396, F128, 31) (dual of [16396, 16332, 32]-code), using
- construction X applied to C([0,15]) ⊂ C([0,13]) [i] based on
- linear OA(12861, 16385, F128, 31) (dual of [16385, 16324, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,15], and minimum distance d ≥ |{−15,−14,…,15}|+1 = 32 (BCH-bound) [i]
- 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(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([0,15]) ⊂ C([0,13]) [i] based on
- discarding parts of the base [i] based on linear OA(12864, 16396, F128, 31) (dual of [16396, 16332, 32]-code), using
- OOA 15-folding and stacking with additional row [i] based on OA(6475, 16396, S64, 31), using
- net defined by OOA [i] based on OOA(6475, 1093, S64, 31, 31), using
(44, 44+30, 7542)-Net over F64 — Digital
Digital (44, 74, 7542)-net over F64, using
(44, 44+30, large)-Net in Base 64 — Upper bound on s
There is no (44, 74, large)-net in base 64, because
- 28 times m-reduction [i] would yield (44, 46, large)-net in base 64, but