Best Known (42−9, 42, s)-Nets in Base 3
(42−9, 42, 546)-Net over F3 — Constructive and digital
Digital (33, 42, 546)-net over F3, using
- net defined by OOA [i] based on linear OOA(342, 546, F3, 9, 9) (dual of [(546, 9), 4872, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(342, 2185, F3, 9) (dual of [2185, 2143, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(342, 2186, F3, 9) (dual of [2186, 2144, 10]-code), using
- 1 times truncation [i] 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]
- 1 times truncation [i] based on linear OA(343, 2187, F3, 10) (dual of [2187, 2144, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(342, 2186, F3, 9) (dual of [2186, 2144, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(342, 2185, F3, 9) (dual of [2185, 2143, 10]-code), using
(42−9, 42, 1093)-Net over F3 — Digital
Digital (33, 42, 1093)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(342, 1093, F3, 2, 9) (dual of [(1093, 2), 2144, 10]-NRT-code), using
- OOA 2-folding [i] based on linear OA(342, 2186, F3, 9) (dual of [2186, 2144, 10]-code), using
- 1 times truncation [i] 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]
- 1 times truncation [i] based on linear OA(343, 2187, F3, 10) (dual of [2187, 2144, 11]-code), using
- OOA 2-folding [i] based on linear OA(342, 2186, F3, 9) (dual of [2186, 2144, 10]-code), using
(42−9, 42, 85999)-Net in Base 3 — Upper bound on s
There is no (33, 42, 86000)-net in base 3, because
- 1 times m-reduction [i] would yield (33, 41, 86000)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 36 473147 450661 648001 > 341 [i]