Best Known (165−30, 165, s)-Nets in Base 8
(165−30, 165, 17478)-Net over F8 — Constructive and digital
Digital (135, 165, 17478)-net over F8, using
- 84 times duplication [i] based on digital (131, 161, 17478)-net over F8, using
- net defined by OOA [i] based on linear OOA(8161, 17478, F8, 30, 30) (dual of [(17478, 30), 524179, 31]-NRT-code), using
- OA 15-folding and stacking [i] based on linear OA(8161, 262170, F8, 30) (dual of [262170, 262009, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(8161, 262172, F8, 30) (dual of [262172, 262011, 31]-code), using
- construction X applied to Ce(29) ⊂ Ce(25) [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(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(84, 28, F8, 3) (dual of [28, 24, 4]-code or 28-cap in PG(3,8)), using
- construction X applied to Ce(29) ⊂ Ce(25) [i] based on
- discarding factors / shortening the dual code based on linear OA(8161, 262172, F8, 30) (dual of [262172, 262011, 31]-code), using
- OA 15-folding and stacking [i] based on linear OA(8161, 262170, F8, 30) (dual of [262170, 262009, 31]-code), using
- net defined by OOA [i] based on linear OOA(8161, 17478, F8, 30, 30) (dual of [(17478, 30), 524179, 31]-NRT-code), using
(165−30, 165, 262184)-Net over F8 — Digital
Digital (135, 165, 262184)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8165, 262184, F8, 30) (dual of [262184, 262019, 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, 262182, F8, 29) (dual of [262182, 262019, 30]-code), using Gilbert–Varšamov bound and bm = 8163 > Vbs−1(k−1) = 79 217073 553732 015872 676134 431291 156495 860742 685896 469236 996028 902956 677292 184156 403809 203841 390482 196172 252257 737789 139508 644821 102401 296843 488916 [i]
- linear OA(80, 2, F8, 0) (dual of [2, 2, 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
(165−30, 165, large)-Net in Base 8 — Upper bound on s
There is no (135, 165, large)-net in base 8, because
- 28 times m-reduction [i] would yield (135, 137, large)-net in base 8, but