Best Known (124, 124+25, s)-Nets in Base 3
(124, 124+25, 1642)-Net over F3 — Constructive and digital
Digital (124, 149, 1642)-net over F3, using
- net defined by OOA [i] based on linear OOA(3149, 1642, F3, 25, 25) (dual of [(1642, 25), 40901, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [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
- OOA 12-folding and stacking with additional row [i] based on linear OA(3149, 19705, F3, 25) (dual of [19705, 19556, 26]-code), using
(124, 124+25, 6959)-Net over F3 — Digital
Digital (124, 149, 6959)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3149, 6959, F3, 2, 25) (dual of [(6959, 2), 13769, 26]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3149, 9852, F3, 2, 25) (dual of [(9852, 2), 19555, 26]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3149, 19704, F3, 25) (dual of [19704, 19555, 26]-code), using
- discarding factors / shortening the dual code 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
- discarding factors / shortening the dual code based on linear OA(3149, 19705, F3, 25) (dual of [19705, 19556, 26]-code), using
- OOA 2-folding [i] based on linear OA(3149, 19704, F3, 25) (dual of [19704, 19555, 26]-code), using
- discarding factors / shortening the dual code based on linear OOA(3149, 9852, F3, 2, 25) (dual of [(9852, 2), 19555, 26]-NRT-code), using
(124, 124+25, 2026863)-Net in Base 3 — Upper bound on s
There is no (124, 149, 2026864)-net in base 3, because
- 1 times m-reduction [i] would yield (124, 148, 2026864)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 41110 036429 347134 265095 914861 439330 002514 437426 757049 909318 862855 833729 > 3148 [i]