Best Known (169−31, 169, s)-Nets in Base 8
(169−31, 169, 17478)-Net over F8 — Constructive and digital
Digital (138, 169, 17478)-net over F8, using
- 82 times duplication [i] based on digital (136, 167, 17478)-net over F8, using
- net defined by OOA [i] based on linear OOA(8167, 17478, F8, 31, 31) (dual of [(17478, 31), 541651, 32]-NRT-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(8167, 262171, F8, 31) (dual of [262171, 262004, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(8167, 262172, F8, 31) (dual of [262172, 262005, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(26) [i] based on
- linear OA(8163, 262144, F8, 31) (dual of [262144, 261981, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(8139, 262144, F8, 27) (dual of [262144, 262005, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(84, 28, F8, 3) (dual of [28, 24, 4]-code or 28-cap in PG(3,8)), using
- construction X applied to Ce(30) ⊂ Ce(26) [i] based on
- discarding factors / shortening the dual code based on linear OA(8167, 262172, F8, 31) (dual of [262172, 262005, 32]-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(8167, 262171, F8, 31) (dual of [262171, 262004, 32]-code), using
- net defined by OOA [i] based on linear OOA(8167, 17478, F8, 31, 31) (dual of [(17478, 31), 541651, 32]-NRT-code), using
(169−31, 169, 262180)-Net over F8 — Digital
Digital (138, 169, 262180)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8169, 262180, F8, 31) (dual of [262180, 262011, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(25) [i] based on
- linear OA(8163, 262144, F8, 31) (dual of [262144, 261981, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(8133, 262144, F8, 26) (dual of [262144, 262011, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(86, 36, F8, 4) (dual of [36, 30, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(86, 56, F8, 4) (dual of [56, 50, 5]-code), using
- 1 times truncation [i] based on linear OA(87, 57, F8, 5) (dual of [57, 50, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(86, 56, F8, 4) (dual of [56, 50, 5]-code), using
- construction X applied to Ce(30) ⊂ Ce(25) [i] based on
(169−31, 169, large)-Net in Base 8 — Upper bound on s
There is no (138, 169, large)-net in base 8, because
- 29 times m-reduction [i] would yield (138, 140, large)-net in base 8, but