Best Known (136, 166, s)-Nets in Base 8
(136, 166, 17478)-Net over F8 — Constructive and digital
Digital (136, 166, 17478)-net over F8, using
- 1 times m-reduction [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
(136, 166, 262186)-Net over F8 — Digital
Digital (136, 166, 262186)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8166, 262186, F8, 30) (dual of [262186, 262020, 31]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(8163, 262180, F8, 30) (dual of [262180, 262017, 31]-code), using
- construction X applied to Ce(29) ⊂ Ce(24) [i] based on
- linear OA(8157, 262144, F8, 30) (dual of [262144, 261987, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(8127, 262144, F8, 25) (dual of [262144, 262017, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [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(29) ⊂ Ce(24) [i] based on
- linear OA(8163, 262183, F8, 29) (dual of [262183, 262020, 30]-code), using Gilbert–Varšamov bound and bm = 8163 > Vbs−1(k−1) = 79 225534 522668 794919 849743 214776 354635 507189 234292 265800 414542 250075 954864 306244 428202 720652 330637 380882 227707 064153 271539 387705 369649 943091 453348 [i]
- linear OA(80, 3, F8, 0) (dual of [3, 3, 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
- linear OA(8163, 262180, F8, 30) (dual of [262180, 262017, 31]-code), using
- construction X with Varšamov bound [i] based on
(136, 166, large)-Net in Base 8 — Upper bound on s
There is no (136, 166, large)-net in base 8, because
- 28 times m-reduction [i] would yield (136, 138, large)-net in base 8, but