Best Known (67, 89, s)-Nets in Base 64
(67, 89, 762600)-Net over F64 — Constructive and digital
Digital (67, 89, 762600)-net over F64, using
- t-expansion [i] based on digital (66, 89, 762600)-net over F64, using
- net defined by OOA [i] based on linear OOA(6489, 762600, F64, 23, 23) (dual of [(762600, 23), 17539711, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(6489, 8388601, F64, 23) (dual of [8388601, 8388512, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(6489, large, F64, 23) (dual of [large, large−89, 24]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 648−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(6489, large, F64, 23) (dual of [large, large−89, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(6489, 8388601, F64, 23) (dual of [8388601, 8388512, 24]-code), using
- net defined by OOA [i] based on linear OOA(6489, 762600, F64, 23, 23) (dual of [(762600, 23), 17539711, 24]-NRT-code), using
(67, 89, large)-Net over F64 — Digital
Digital (67, 89, large)-net over F64, using
- 641 times duplication [i] based on digital (66, 88, large)-net over F64, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(6488, large, F64, 22) (dual of [large, large−88, 23]-code), using
- 3 times code embedding in larger space [i] based on linear OA(6485, large, F64, 22) (dual of [large, large−85, 23]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 644−1, defining interval I = [0,21], and designed minimum distance d ≥ |I|+1 = 23 [i]
- 3 times code embedding in larger space [i] based on linear OA(6485, large, F64, 22) (dual of [large, large−85, 23]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(6488, large, F64, 22) (dual of [large, large−88, 23]-code), using
(67, 89, large)-Net in Base 64 — Upper bound on s
There is no (67, 89, large)-net in base 64, because
- 20 times m-reduction [i] would yield (67, 69, large)-net in base 64, but