Best Known (109−23, 109, s)-Nets in Base 8
(109−23, 109, 2982)-Net over F8 — Constructive and digital
Digital (86, 109, 2982)-net over F8, using
- 81 times duplication [i] based on digital (85, 108, 2982)-net over F8, using
- net defined by OOA [i] based on linear OOA(8108, 2982, F8, 23, 23) (dual of [(2982, 23), 68478, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(8108, 32803, F8, 23) (dual of [32803, 32695, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(8108, 32806, F8, 23) (dual of [32806, 32698, 24]-code), using
- construction X applied to C([0,11]) ⊂ C([0,8]) [i] based on
- linear OA(8101, 32769, F8, 23) (dual of [32769, 32668, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 810−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- linear OA(871, 32769, F8, 17) (dual of [32769, 32698, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 810−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(87, 37, F8, 5) (dual of [37, 30, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(87, 57, F8, 5) (dual of [57, 50, 6]-code), using
- construction X applied to C([0,11]) ⊂ C([0,8]) [i] based on
- discarding factors / shortening the dual code based on linear OA(8108, 32806, F8, 23) (dual of [32806, 32698, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(8108, 32803, F8, 23) (dual of [32803, 32695, 24]-code), using
- net defined by OOA [i] based on linear OOA(8108, 2982, F8, 23, 23) (dual of [(2982, 23), 68478, 24]-NRT-code), using
(109−23, 109, 38572)-Net over F8 — Digital
Digital (86, 109, 38572)-net over F8, using
(109−23, 109, large)-Net in Base 8 — Upper bound on s
There is no (86, 109, large)-net in base 8, because
- 21 times m-reduction [i] would yield (86, 88, large)-net in base 8, but