Best Known (67, 67+15, s)-Nets in Base 8
(67, 67+15, 37452)-Net over F8 — Constructive and digital
Digital (67, 82, 37452)-net over F8, using
- net defined by OOA [i] based on linear OOA(882, 37452, F8, 15, 15) (dual of [(37452, 15), 561698, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(882, 262165, F8, 15) (dual of [262165, 262083, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(11) [i] based on
- linear OA(879, 262144, F8, 15) (dual of [262144, 262065, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [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(83, 21, F8, 2) (dual of [21, 18, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(83, 63, F8, 2) (dual of [63, 60, 3]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 82−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- discarding factors / shortening the dual code based on linear OA(83, 63, F8, 2) (dual of [63, 60, 3]-code), using
- construction X applied to Ce(14) ⊂ Ce(11) [i] based on
- OOA 7-folding and stacking with additional row [i] based on linear OA(882, 262165, F8, 15) (dual of [262165, 262083, 16]-code), using
(67, 67+15, 262165)-Net over F8 — Digital
Digital (67, 82, 262165)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(882, 262165, F8, 15) (dual of [262165, 262083, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(11) [i] based on
- linear OA(879, 262144, F8, 15) (dual of [262144, 262065, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [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(83, 21, F8, 2) (dual of [21, 18, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(83, 63, F8, 2) (dual of [63, 60, 3]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 82−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- discarding factors / shortening the dual code based on linear OA(83, 63, F8, 2) (dual of [63, 60, 3]-code), using
- construction X applied to Ce(14) ⊂ Ce(11) [i] based on
(67, 67+15, large)-Net in Base 8 — Upper bound on s
There is no (67, 82, large)-net in base 8, because
- 13 times m-reduction [i] would yield (67, 69, large)-net in base 8, but