Best Known (105, 105+21, s)-Nets in Base 3
(105, 105+21, 1968)-Net over F3 — Constructive and digital
Digital (105, 126, 1968)-net over F3, using
- net defined by OOA [i] based on linear OOA(3126, 1968, F3, 21, 21) (dual of [(1968, 21), 41202, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(3126, 19681, F3, 21) (dual of [19681, 19555, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(3126, 19682, F3, 21) (dual of [19682, 19556, 22]-code), using
- 1 times truncation [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]
- 1 times truncation [i] based on linear OA(3127, 19683, F3, 22) (dual of [19683, 19556, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(3126, 19682, F3, 21) (dual of [19682, 19556, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(3126, 19681, F3, 21) (dual of [19681, 19555, 22]-code), using
(105, 105+21, 7293)-Net over F3 — Digital
Digital (105, 126, 7293)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3126, 7293, F3, 2, 21) (dual of [(7293, 2), 14460, 22]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3126, 9841, F3, 2, 21) (dual of [(9841, 2), 19556, 22]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3126, 19682, F3, 21) (dual of [19682, 19556, 22]-code), using
- 1 times truncation [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]
- 1 times truncation [i] based on linear OA(3127, 19683, F3, 22) (dual of [19683, 19556, 23]-code), using
- OOA 2-folding [i] based on linear OA(3126, 19682, F3, 21) (dual of [19682, 19556, 22]-code), using
- discarding factors / shortening the dual code based on linear OOA(3126, 9841, F3, 2, 21) (dual of [(9841, 2), 19556, 22]-NRT-code), using
(105, 105+21, 2084298)-Net in Base 3 — Upper bound on s
There is no (105, 126, 2084299)-net in base 3, because
- 1 times m-reduction [i] would yield (105, 125, 2084299)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 436673 583321 062985 512943 194462 727890 303621 225909 978152 877541 > 3125 [i]