Best Known (136−22, 136, s)-Nets in Base 3
(136−22, 136, 1792)-Net over F3 — Constructive and digital
Digital (114, 136, 1792)-net over F3, using
- 31 times duplication [i] based on digital (113, 135, 1792)-net over F3, using
- net defined by OOA [i] based on linear OOA(3135, 1792, F3, 22, 22) (dual of [(1792, 22), 39289, 23]-NRT-code), using
- OA 11-folding and stacking [i] based on linear OA(3135, 19712, F3, 22) (dual of [19712, 19577, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(3135, 19718, F3, 22) (dual of [19718, 19583, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(16) [i] based on
- linear OA(3127, 19683, F3, 22) (dual of [19683, 19556, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(3100, 19683, F3, 17) (dual of [19683, 19583, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(38, 35, F3, 4) (dual of [35, 27, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- the narrow-sense BCH-code C(I) with length 41 | 38−1, defining interval I = [1,1], and minimum distance d ≥ |{−3,−1,1,3}|+1 = 5 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- construction X applied to Ce(21) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(3135, 19718, F3, 22) (dual of [19718, 19583, 23]-code), using
- OA 11-folding and stacking [i] based on linear OA(3135, 19712, F3, 22) (dual of [19712, 19577, 23]-code), using
- net defined by OOA [i] based on linear OOA(3135, 1792, F3, 22, 22) (dual of [(1792, 22), 39289, 23]-NRT-code), using
(136−22, 136, 9168)-Net over F3 — Digital
Digital (114, 136, 9168)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3136, 9168, F3, 2, 22) (dual of [(9168, 2), 18200, 23]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3136, 9859, F3, 2, 22) (dual of [(9859, 2), 19582, 23]-NRT-code), using
- 31 times duplication [i] based on linear OOA(3135, 9859, F3, 2, 22) (dual of [(9859, 2), 19583, 23]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3135, 19718, F3, 22) (dual of [19718, 19583, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(16) [i] based on
- linear OA(3127, 19683, F3, 22) (dual of [19683, 19556, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(3100, 19683, F3, 17) (dual of [19683, 19583, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(38, 35, F3, 4) (dual of [35, 27, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- the narrow-sense BCH-code C(I) with length 41 | 38−1, defining interval I = [1,1], and minimum distance d ≥ |{−3,−1,1,3}|+1 = 5 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- construction X applied to Ce(21) ⊂ Ce(16) [i] based on
- OOA 2-folding [i] based on linear OA(3135, 19718, F3, 22) (dual of [19718, 19583, 23]-code), using
- 31 times duplication [i] based on linear OOA(3135, 9859, F3, 2, 22) (dual of [(9859, 2), 19583, 23]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3136, 9859, F3, 2, 22) (dual of [(9859, 2), 19582, 23]-NRT-code), using
(136−22, 136, 1945071)-Net in Base 3 — Upper bound on s
There is no (114, 136, 1945072)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 77355 736563 772252 289369 984394 335298 092725 825466 698260 333813 401025 > 3136 [i]