Best Known (125−15, 125, s)-Nets in Base 9
(125−15, 125, 2396772)-Net over F9 — Constructive and digital
Digital (110, 125, 2396772)-net over F9, using
- (u, u+v)-construction [i] based on
- digital (4, 11, 30)-net over F9, using
- net from sequence [i] based on digital (4, 29)-sequence over F9, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 4 and N(F) ≥ 30, using
- net from sequence [i] based on digital (4, 29)-sequence over F9, using
- digital (99, 114, 2396742)-net over F9, using
- trace code for nets [i] based on digital (42, 57, 1198371)-net over F81, using
- net defined by OOA [i] based on linear OOA(8157, 1198371, F81, 15, 15) (dual of [(1198371, 15), 17975508, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(8157, 8388598, F81, 15) (dual of [8388598, 8388541, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(8157, large, F81, 15) (dual of [large, large−57, 16]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523361 | 818−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(8157, large, F81, 15) (dual of [large, large−57, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(8157, 8388598, F81, 15) (dual of [8388598, 8388541, 16]-code), using
- net defined by OOA [i] based on linear OOA(8157, 1198371, F81, 15, 15) (dual of [(1198371, 15), 17975508, 16]-NRT-code), using
- trace code for nets [i] based on digital (42, 57, 1198371)-net over F81, using
- digital (4, 11, 30)-net over F9, using
(125−15, 125, large)-Net over F9 — Digital
Digital (110, 125, large)-net over F9, using
- 4 times m-reduction [i] based on digital (110, 129, large)-net over F9, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(9129, large, F9, 19) (dual of [large, large−129, 20]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523361 | 916−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(9129, large, F9, 19) (dual of [large, large−129, 20]-code), using
(125−15, 125, large)-Net in Base 9 — Upper bound on s
There is no (110, 125, large)-net in base 9, because
- 13 times m-reduction [i] would yield (110, 112, large)-net in base 9, but