Best Known (33, 33+17, s)-Nets in Base 64
(33, 33+17, 32768)-Net over F64 — Constructive and digital
Digital (33, 50, 32768)-net over F64, using
- 641 times duplication [i] based on 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
(33, 33+17, 131076)-Net over F64 — Digital
Digital (33, 50, 131076)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6450, 131076, F64, 2, 17) (dual of [(131076, 2), 262102, 18]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6450, 262152, F64, 17) (dual of [262152, 262102, 18]-code), using
- construction X applied to C([0,8]) ⊂ C([0,7]) [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]
- linear OA(6443, 262145, F64, 15) (dual of [262145, 262102, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 646−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(641, 7, F64, 1) (dual of [7, 6, 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([0,8]) ⊂ C([0,7]) [i] based on
- OOA 2-folding [i] based on linear OA(6450, 262152, F64, 17) (dual of [262152, 262102, 18]-code), using
(33, 33+17, large)-Net in Base 64 — Upper bound on s
There is no (33, 50, large)-net in base 64, because
- 15 times m-reduction [i] would yield (33, 35, large)-net in base 64, but