Best Known (148−24, 148, s)-Nets in Base 3
(148−24, 148, 1642)-Net over F3 — Constructive and digital
Digital (124, 148, 1642)-net over F3, using
- net defined by OOA [i] based on linear OOA(3148, 1642, F3, 24, 24) (dual of [(1642, 24), 39260, 25]-NRT-code), using
- OA 12-folding and stacking [i] based on linear OA(3148, 19704, F3, 24) (dual of [19704, 19556, 25]-code), using
- 1 times truncation [i] based on linear OA(3149, 19705, F3, 25) (dual of [19705, 19556, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(21) [i] based on
- linear OA(3145, 19683, F3, 25) (dual of [19683, 19538, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [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(34, 22, F3, 2) (dual of [22, 18, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- Hamming code H(4,3) [i]
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- construction X applied to Ce(24) ⊂ Ce(21) [i] based on
- 1 times truncation [i] based on linear OA(3149, 19705, F3, 25) (dual of [19705, 19556, 26]-code), using
- OA 12-folding and stacking [i] based on linear OA(3148, 19704, F3, 24) (dual of [19704, 19556, 25]-code), using
(148−24, 148, 8987)-Net over F3 — Digital
Digital (124, 148, 8987)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3148, 8987, F3, 2, 24) (dual of [(8987, 2), 17826, 25]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3148, 9852, F3, 2, 24) (dual of [(9852, 2), 19556, 25]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(3146, 9851, F3, 2, 24) (dual of [(9851, 2), 19556, 25]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3146, 19702, F3, 24) (dual of [19702, 19556, 25]-code), using
- construction X applied to Ce(24) ⊂ Ce(21) [i] based on
- linear OA(3145, 19683, F3, 25) (dual of [19683, 19538, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [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(31, 19, F3, 1) (dual of [19, 18, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(24) ⊂ Ce(21) [i] based on
- OOA 2-folding [i] based on linear OA(3146, 19702, F3, 24) (dual of [19702, 19556, 25]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(3146, 9851, F3, 2, 24) (dual of [(9851, 2), 19556, 25]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3148, 9852, F3, 2, 24) (dual of [(9852, 2), 19556, 25]-NRT-code), using
(148−24, 148, 2026863)-Net in Base 3 — Upper bound on s
There is no (124, 148, 2026864)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 41110 036429 347134 265095 914861 439330 002514 437426 757049 909318 862855 833729 > 3148 [i]