Best Known (187, 187+15, s)-Nets in Base 3
(187, 187+15, 1375523)-Net over F3 — Constructive and digital
Digital (187, 202, 1375523)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (45, 52, 177152)-net over F3, using
- net defined by OOA [i] based on linear OOA(352, 177152, F3, 7, 7) (dual of [(177152, 7), 1240012, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(352, 531457, F3, 7) (dual of [531457, 531405, 8]-code), using
- 2 times code embedding in larger space [i] based on linear OA(350, 531455, F3, 7) (dual of [531455, 531405, 8]-code), using
- construction X4 applied to Ce(6) ⊂ Ce(4) [i] based on
- linear OA(349, 531441, F3, 7) (dual of [531441, 531392, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(337, 531441, F3, 5) (dual of [531441, 531404, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(313, 14, F3, 13) (dual of [14, 1, 14]-code or 14-arc in PG(12,3)), using
- dual of repetition code with length 14 [i]
- linear OA(31, 14, F3, 1) (dual of [14, 13, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X4 applied to Ce(6) ⊂ Ce(4) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(350, 531455, F3, 7) (dual of [531455, 531405, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(352, 531457, F3, 7) (dual of [531457, 531405, 8]-code), using
- net defined by OOA [i] based on linear OOA(352, 177152, F3, 7, 7) (dual of [(177152, 7), 1240012, 8]-NRT-code), 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 (45, 52, 177152)-net over F3, using
(187, 187+15, large)-Net over F3 — Digital
Digital (187, 202, large)-net over F3, using
- t-expansion [i] based on digital (186, 202, large)-net over F3, using
- 1 times m-reduction [i] based on digital (186, 203, large)-net over F3, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3203, large, F3, 17) (dual of [large, large−203, 18]-code), using
- 37 times code embedding in larger space [i] based on linear OA(3166, large, F3, 17) (dual of [large, large−166, 18]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 14348906 = 315−1, defining interval I = [0,16], and designed minimum distance d ≥ |I|+1 = 18 [i]
- 37 times code embedding in larger space [i] based on linear OA(3166, large, F3, 17) (dual of [large, large−166, 18]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3203, large, F3, 17) (dual of [large, large−203, 18]-code), using
- 1 times m-reduction [i] based on digital (186, 203, large)-net over F3, using
(187, 187+15, large)-Net in Base 3 — Upper bound on s
There is no (187, 202, large)-net in base 3, because
- 13 times m-reduction [i] would yield (187, 189, large)-net in base 3, but