Best Known (157−15, 157, s)-Nets in Base 3
(157−15, 157, 1198375)-Net over F3 — Constructive and digital
Digital (142, 157, 1198375)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (0, 7, 4)-net over F3, using
- net from sequence [i] based on digital (0, 3)-sequence over F3, using
- Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 0 and N(F) ≥ 4, using
- the rational function field F3(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 3)-sequence over F3, using
- digital (135, 150, 1198371)-net over F3, using
- net defined by OOA [i] based on linear OOA(3150, 1198371, F3, 15, 15) (dual of [(1198371, 15), 17975415, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(3150, 8388598, F3, 15) (dual of [8388598, 8388448, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(3150, large, F3, 15) (dual of [large, large−150, 16]-code), using
- the primitive narrow-sense BCH-code C(I) with length 14348906 = 315−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- discarding factors / shortening the dual code based on linear OA(3150, large, F3, 15) (dual of [large, large−150, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(3150, 8388598, F3, 15) (dual of [8388598, 8388448, 16]-code), using
- net defined by OOA [i] based on linear OOA(3150, 1198371, F3, 15, 15) (dual of [(1198371, 15), 17975415, 16]-NRT-code), using
- digital (0, 7, 4)-net over F3, using
(157−15, 157, 3847214)-Net over F3 — Digital
Digital (142, 157, 3847214)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3157, 3847214, F3, 2, 15) (dual of [(3847214, 2), 7694271, 16]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3157, 4194305, F3, 2, 15) (dual of [(4194305, 2), 8388453, 16]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(37, 4, F3, 2, 7) (dual of [(4, 2), 1, 8]-NRT-code), using
- dual of repetition NRT-code with length 4 [i]
- linear OOA(3150, 4194301, F3, 2, 15) (dual of [(4194301, 2), 8388452, 16]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3150, 8388602, F3, 15) (dual of [8388602, 8388452, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(3150, large, F3, 15) (dual of [large, large−150, 16]-code), using
- the primitive narrow-sense BCH-code C(I) with length 14348906 = 315−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- discarding factors / shortening the dual code based on linear OA(3150, large, F3, 15) (dual of [large, large−150, 16]-code), using
- OOA 2-folding [i] based on linear OA(3150, 8388602, F3, 15) (dual of [8388602, 8388452, 16]-code), using
- linear OOA(37, 4, F3, 2, 7) (dual of [(4, 2), 1, 8]-NRT-code), using
- (u, u+v)-construction [i] based on
- discarding factors / shortening the dual code based on linear OOA(3157, 4194305, F3, 2, 15) (dual of [(4194305, 2), 8388453, 16]-NRT-code), using
(157−15, 157, large)-Net in Base 3 — Upper bound on s
There is no (142, 157, large)-net in base 3, because
- 13 times m-reduction [i] would yield (142, 144, large)-net in base 3, but