Best Known (51, 65, s)-Nets in Base 8
(51, 65, 4684)-Net over F8 — Constructive and digital
Digital (51, 65, 4684)-net over F8, using
- net defined by OOA [i] based on linear OOA(865, 4684, F8, 14, 14) (dual of [(4684, 14), 65511, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(865, 32788, F8, 14) (dual of [32788, 32723, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(865, 32792, F8, 14) (dual of [32792, 32727, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(9) [i] based on
- linear OA(861, 32768, F8, 14) (dual of [32768, 32707, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(841, 32768, F8, 10) (dual of [32768, 32727, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(84, 24, F8, 3) (dual of [24, 20, 4]-code or 24-cap in PG(3,8)), using
- construction X applied to Ce(13) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(865, 32792, F8, 14) (dual of [32792, 32727, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(865, 32788, F8, 14) (dual of [32788, 32723, 15]-code), using
(51, 65, 32792)-Net over F8 — Digital
Digital (51, 65, 32792)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(865, 32792, F8, 14) (dual of [32792, 32727, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(9) [i] based on
- linear OA(861, 32768, F8, 14) (dual of [32768, 32707, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(841, 32768, F8, 10) (dual of [32768, 32727, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(84, 24, F8, 3) (dual of [24, 20, 4]-code or 24-cap in PG(3,8)), using
- construction X applied to Ce(13) ⊂ Ce(9) [i] based on
(51, 65, large)-Net in Base 8 — Upper bound on s
There is no (51, 65, large)-net in base 8, because
- 12 times m-reduction [i] would yield (51, 53, large)-net in base 8, but