Best Known (67−13, 67, s)-Nets in Base 8
(67−13, 67, 43691)-Net over F8 — Constructive and digital
Digital (54, 67, 43691)-net over F8, using
- net defined by OOA [i] based on linear OOA(867, 43691, F8, 13, 13) (dual of [(43691, 13), 567916, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(867, 262147, F8, 13) (dual of [262147, 262080, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(867, 262150, F8, 13) (dual of [262150, 262083, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- linear OA(867, 262144, F8, 13) (dual of [262144, 262077, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(861, 262144, F8, 12) (dual of [262144, 262083, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(80, 6, F8, 0) (dual of [6, 6, 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(12) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(867, 262150, F8, 13) (dual of [262150, 262083, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(867, 262147, F8, 13) (dual of [262147, 262080, 14]-code), using
(67−13, 67, 183841)-Net over F8 — Digital
Digital (54, 67, 183841)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(867, 183841, F8, 13) (dual of [183841, 183774, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(867, 262144, F8, 13) (dual of [262144, 262077, 14]-code), using
- an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- discarding factors / shortening the dual code based on linear OA(867, 262144, F8, 13) (dual of [262144, 262077, 14]-code), using
(67−13, 67, large)-Net in Base 8 — Upper bound on s
There is no (54, 67, large)-net in base 8, because
- 11 times m-reduction [i] would yield (54, 56, large)-net in base 8, but