Best Known (33, 33+10, s)-Nets in Base 3
(33, 33+10, 437)-Net over F3 — Constructive and digital
Digital (33, 43, 437)-net over F3, using
- net defined by OOA [i] based on linear OOA(343, 437, F3, 10, 10) (dual of [(437, 10), 4327, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(343, 2185, F3, 10) (dual of [2185, 2142, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(343, 2187, F3, 10) (dual of [2187, 2144, 11]-code), using
- an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- discarding factors / shortening the dual code based on linear OA(343, 2187, F3, 10) (dual of [2187, 2144, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(343, 2185, F3, 10) (dual of [2185, 2142, 11]-code), using
(33, 33+10, 1047)-Net over F3 — Digital
Digital (33, 43, 1047)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(343, 1047, F3, 2, 10) (dual of [(1047, 2), 2051, 11]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(343, 1093, F3, 2, 10) (dual of [(1093, 2), 2143, 11]-NRT-code), using
- OOA 2-folding [i] based on linear OA(343, 2186, F3, 10) (dual of [2186, 2143, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(343, 2187, F3, 10) (dual of [2187, 2144, 11]-code), using
- an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- discarding factors / shortening the dual code based on linear OA(343, 2187, F3, 10) (dual of [2187, 2144, 11]-code), using
- OOA 2-folding [i] based on linear OA(343, 2186, F3, 10) (dual of [2186, 2143, 11]-code), using
- discarding factors / shortening the dual code based on linear OOA(343, 1093, F3, 2, 10) (dual of [(1093, 2), 2143, 11]-NRT-code), using
(33, 33+10, 16516)-Net in Base 3 — Upper bound on s
There is no (33, 43, 16517)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 328 258543 533148 081179 > 343 [i]