Best Known (64−17, 64, s)-Nets in Base 64
(64−17, 64, 33792)-Net over F64 — Constructive and digital
Digital (47, 64, 33792)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (7, 15, 1024)-net over F64, using
- net defined by OOA [i] based on linear OOA(6415, 1024, F64, 8, 8) (dual of [(1024, 8), 8177, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(6415, 4096, F64, 8) (dual of [4096, 4081, 9]-code), using
- an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- OA 4-folding and stacking [i] based on linear OA(6415, 4096, F64, 8) (dual of [4096, 4081, 9]-code), using
- net defined by OOA [i] based on linear OOA(6415, 1024, F64, 8, 8) (dual of [(1024, 8), 8177, 9]-NRT-code), using
- digital (32, 49, 32768)-net over F64, using
- net defined by OOA [i] based on linear OOA(6449, 32768, F64, 17, 17) (dual of [(32768, 17), 557007, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(6449, 262145, F64, 17) (dual of [262145, 262096, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 262145 | 646−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- OOA 8-folding and stacking with additional row [i] based on linear OA(6449, 262145, F64, 17) (dual of [262145, 262096, 18]-code), using
- net defined by OOA [i] based on linear OOA(6449, 32768, F64, 17, 17) (dual of [(32768, 17), 557007, 18]-NRT-code), using
- digital (7, 15, 1024)-net over F64, using
(64−17, 64, 262147)-Net in Base 64 — Constructive
(47, 64, 262147)-net in base 64, using
- net defined by OOA [i] based on OOA(6464, 262147, S64, 17, 17), using
- OOA 8-folding and stacking with additional row [i] based on OA(6464, 2097177, S64, 17), using
- 1 times code embedding in larger space [i] based on OA(6463, 2097176, S64, 17), using
- discarding parts of the base [i] based on linear OA(12854, 2097176, F128, 17) (dual of [2097176, 2097122, 18]-code), using
- construction X applied to C([0,8]) ⊂ C([0,5]) [i] based on
- linear OA(12849, 2097153, F128, 17) (dual of [2097153, 2097104, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(12831, 2097153, F128, 11) (dual of [2097153, 2097122, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(1285, 23, F128, 5) (dual of [23, 18, 6]-code or 23-arc in PG(4,128)), using
- discarding factors / shortening the dual code based on linear OA(1285, 128, F128, 5) (dual of [128, 123, 6]-code or 128-arc in PG(4,128)), using
- Reed–Solomon code RS(123,128) [i]
- discarding factors / shortening the dual code based on linear OA(1285, 128, F128, 5) (dual of [128, 123, 6]-code or 128-arc in PG(4,128)), using
- construction X applied to C([0,8]) ⊂ C([0,5]) [i] based on
- discarding parts of the base [i] based on linear OA(12854, 2097176, F128, 17) (dual of [2097176, 2097122, 18]-code), using
- 1 times code embedding in larger space [i] based on OA(6463, 2097176, S64, 17), using
- OOA 8-folding and stacking with additional row [i] based on OA(6464, 2097177, S64, 17), using
(64−17, 64, 1811006)-Net over F64 — Digital
Digital (47, 64, 1811006)-net over F64, using
(64−17, 64, large)-Net in Base 64 — Upper bound on s
There is no (47, 64, large)-net in base 64, because
- 15 times m-reduction [i] would yield (47, 49, large)-net in base 64, but