Best Known (51, 51+15, s)-Nets in Base 8
(51, 51+15, 4681)-Net over F8 — Constructive and digital
Digital (51, 66, 4681)-net over F8, using
- net defined by OOA [i] based on linear OOA(866, 4681, F8, 15, 15) (dual of [(4681, 15), 70149, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(866, 32768, F8, 15) (dual of [32768, 32702, 16]-code), using
- an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- OOA 7-folding and stacking with additional row [i] based on linear OA(866, 32768, F8, 15) (dual of [32768, 32702, 16]-code), using
(51, 51+15, 26524)-Net over F8 — Digital
Digital (51, 66, 26524)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(866, 26524, F8, 15) (dual of [26524, 26458, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(866, 32768, F8, 15) (dual of [32768, 32702, 16]-code), using
- an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- discarding factors / shortening the dual code based on linear OA(866, 32768, F8, 15) (dual of [32768, 32702, 16]-code), using
(51, 51+15, large)-Net in Base 8 — Upper bound on s
There is no (51, 66, large)-net in base 8, because
- 13 times m-reduction [i] would yield (51, 53, large)-net in base 8, but