Best Known (78, 78+23, s)-Nets in Base 8
(78, 78+23, 2979)-Net over F8 — Constructive and digital
Digital (78, 101, 2979)-net over F8, using
- net defined by OOA [i] based on linear OOA(8101, 2979, F8, 23, 23) (dual of [(2979, 23), 68416, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(8101, 32770, F8, 23) (dual of [32770, 32669, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(8101, 32773, F8, 23) (dual of [32773, 32672, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(21) [i] based on
- linear OA(8101, 32768, F8, 23) (dual of [32768, 32667, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(896, 32768, F8, 22) (dual of [32768, 32672, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(80, 5, F8, 0) (dual of [5, 5, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(80, s, F8, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(22) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(8101, 32773, F8, 23) (dual of [32773, 32672, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(8101, 32770, F8, 23) (dual of [32770, 32669, 24]-code), using
(78, 78+23, 24752)-Net over F8 — Digital
Digital (78, 101, 24752)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8101, 24752, F8, 23) (dual of [24752, 24651, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(8101, 32768, F8, 23) (dual of [32768, 32667, 24]-code), using
- an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- discarding factors / shortening the dual code based on linear OA(8101, 32768, F8, 23) (dual of [32768, 32667, 24]-code), using
(78, 78+23, large)-Net in Base 8 — Upper bound on s
There is no (78, 101, large)-net in base 8, because
- 21 times m-reduction [i] would yield (78, 80, large)-net in base 8, but