Best Known (20, 33, s)-Nets in Base 64
(20, 33, 763)-Net over F64 — Constructive and digital
Digital (20, 33, 763)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (1, 7, 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 (13, 26, 683)-net over F64, using
- net defined by OOA [i] based on linear OOA(6426, 683, F64, 13, 13) (dual of [(683, 13), 8853, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(6426, 4099, F64, 13) (dual of [4099, 4073, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(6426, 4102, F64, 13) (dual of [4102, 4076, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([0,5]) [i] based on
- linear OA(6425, 4097, F64, 13) (dual of [4097, 4072, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(6421, 4097, F64, 11) (dual of [4097, 4076, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(641, 5, F64, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(641, s, F64, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,6]) ⊂ C([0,5]) [i] based on
- discarding factors / shortening the dual code based on linear OA(6426, 4102, F64, 13) (dual of [4102, 4076, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(6426, 4099, F64, 13) (dual of [4099, 4073, 14]-code), using
- net defined by OOA [i] based on linear OOA(6426, 683, F64, 13, 13) (dual of [(683, 13), 8853, 14]-NRT-code), using
- digital (1, 7, 80)-net over F64, using
(20, 33, 2732)-Net in Base 64 — Constructive
(20, 33, 2732)-net in base 64, using
- net defined by OOA [i] based on OOA(6433, 2732, S64, 13, 13), using
- OOA 6-folding and stacking with additional row [i] based on OA(6433, 16393, S64, 13), using
- discarding factors based on OA(6433, 16396, S64, 13), using
- discarding parts of the base [i] based on linear OA(12828, 16396, F128, 13) (dual of [16396, 16368, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([0,4]) [i] based on
- linear OA(12825, 16385, F128, 13) (dual of [16385, 16360, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(12817, 16385, F128, 9) (dual of [16385, 16368, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (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([0,6]) ⊂ C([0,4]) [i] based on
- discarding parts of the base [i] based on linear OA(12828, 16396, F128, 13) (dual of [16396, 16368, 14]-code), using
- discarding factors based on OA(6433, 16396, S64, 13), using
- OOA 6-folding and stacking with additional row [i] based on OA(6433, 16393, S64, 13), using
(20, 33, 7787)-Net over F64 — Digital
Digital (20, 33, 7787)-net over F64, using
(20, 33, large)-Net in Base 64 — Upper bound on s
There is no (20, 33, large)-net in base 64, because
- 11 times m-reduction [i] would yield (20, 22, large)-net in base 64, but