Best Known (137, 137+26, s)-Nets in Base 3
(137, 137+26, 1516)-Net over F3 — Constructive and digital
Digital (137, 163, 1516)-net over F3, using
- 33 times duplication [i] based on digital (134, 160, 1516)-net over F3, using
- net defined by OOA [i] based on linear OOA(3160, 1516, F3, 26, 26) (dual of [(1516, 26), 39256, 27]-NRT-code), using
- OA 13-folding and stacking [i] based on linear OA(3160, 19708, F3, 26) (dual of [19708, 19548, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(3160, 19716, F3, 26) (dual of [19716, 19556, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(21) [i] based on
- linear OA(3154, 19683, F3, 26) (dual of [19683, 19529, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- 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(36, 33, F3, 3) (dual of [33, 27, 4]-code or 33-cap in PG(5,3)), using
- discarding factors / shortening the dual code based on linear OA(36, 48, F3, 3) (dual of [48, 42, 4]-code or 48-cap in PG(5,3)), using
- construction X applied to Ce(25) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(3160, 19716, F3, 26) (dual of [19716, 19556, 27]-code), using
- OA 13-folding and stacking [i] based on linear OA(3160, 19708, F3, 26) (dual of [19708, 19548, 27]-code), using
- net defined by OOA [i] based on linear OOA(3160, 1516, F3, 26, 26) (dual of [(1516, 26), 39256, 27]-NRT-code), using
(137, 137+26, 9859)-Net over F3 — Digital
Digital (137, 163, 9859)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3163, 9859, F3, 2, 26) (dual of [(9859, 2), 19555, 27]-NRT-code), using
- 31 times duplication [i] based on linear OOA(3162, 9859, F3, 2, 26) (dual of [(9859, 2), 19556, 27]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(3160, 9858, F3, 2, 26) (dual of [(9858, 2), 19556, 27]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3160, 19716, F3, 26) (dual of [19716, 19556, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(21) [i] based on
- linear OA(3154, 19683, F3, 26) (dual of [19683, 19529, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- 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(36, 33, F3, 3) (dual of [33, 27, 4]-code or 33-cap in PG(5,3)), using
- discarding factors / shortening the dual code based on linear OA(36, 48, F3, 3) (dual of [48, 42, 4]-code or 48-cap in PG(5,3)), using
- construction X applied to Ce(25) ⊂ Ce(21) [i] based on
- OOA 2-folding [i] based on linear OA(3160, 19716, F3, 26) (dual of [19716, 19556, 27]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(3160, 9858, F3, 2, 26) (dual of [(9858, 2), 19556, 27]-NRT-code), using
- 31 times duplication [i] based on linear OOA(3162, 9859, F3, 2, 26) (dual of [(9859, 2), 19556, 27]-NRT-code), using
(137, 137+26, 2721076)-Net in Base 3 — Upper bound on s
There is no (137, 163, 2721077)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 589883 667395 322633 523221 510398 664241 263793 427038 595384 604568 454923 387768 443739 > 3163 [i]