Best Known (65, 80, s)-Nets in Base 8
(65, 80, 37450)-Net over F8 — Constructive and digital
Digital (65, 80, 37450)-net over F8, using
- net defined by OOA [i] based on linear OOA(880, 37450, F8, 15, 15) (dual of [(37450, 15), 561670, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(880, 262151, F8, 15) (dual of [262151, 262071, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(880, 262157, F8, 15) (dual of [262157, 262077, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(12) [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(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(81, 13, F8, 1) (dual of [13, 12, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(81, s, F8, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(14) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(880, 262157, F8, 15) (dual of [262157, 262077, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(880, 262151, F8, 15) (dual of [262151, 262071, 16]-code), using
(65, 80, 249060)-Net over F8 — Digital
Digital (65, 80, 249060)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(880, 249060, F8, 15) (dual of [249060, 248980, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(880, 262157, F8, 15) (dual of [262157, 262077, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(12) [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(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(81, 13, F8, 1) (dual of [13, 12, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(81, s, F8, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(14) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(880, 262157, F8, 15) (dual of [262157, 262077, 16]-code), using
(65, 80, large)-Net in Base 8 — Upper bound on s
There is no (65, 80, large)-net in base 8, because
- 13 times m-reduction [i] would yield (65, 67, large)-net in base 8, but