Best Known (123, 140, s)-Nets in Base 9
(123, 140, 2097170)-Net over F9 — Constructive and digital
Digital (123, 140, 2097170)-net over F9, using
- (u, u+v)-construction [i] based on
- digital (2, 10, 20)-net over F9, using
- net from sequence [i] based on digital (2, 19)-sequence over F9, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 2 and N(F) ≥ 20, using
- net from sequence [i] based on digital (2, 19)-sequence over F9, using
- digital (113, 130, 2097150)-net over F9, using
- net defined by OOA [i] based on linear OOA(9130, 2097150, F9, 18, 17) (dual of [(2097150, 18), 37748570, 18]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OOA(9130, 8388601, F9, 2, 17) (dual of [(8388601, 2), 16777072, 18]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(9130, 8388602, F9, 2, 17) (dual of [(8388602, 2), 16777074, 18]-NRT-code), using
- trace code [i] based on linear OOA(8165, 4194301, F81, 2, 17) (dual of [(4194301, 2), 8388537, 18]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8165, 8388602, F81, 17) (dual of [8388602, 8388537, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(8165, large, F81, 17) (dual of [large, large−65, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523361 | 818−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(8165, large, F81, 17) (dual of [large, large−65, 18]-code), using
- OOA 2-folding [i] based on linear OA(8165, 8388602, F81, 17) (dual of [8388602, 8388537, 18]-code), using
- trace code [i] based on linear OOA(8165, 4194301, F81, 2, 17) (dual of [(4194301, 2), 8388537, 18]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(9130, 8388602, F9, 2, 17) (dual of [(8388602, 2), 16777074, 18]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OOA(9130, 8388601, F9, 2, 17) (dual of [(8388601, 2), 16777072, 18]-NRT-code), using
- net defined by OOA [i] based on linear OOA(9130, 2097150, F9, 18, 17) (dual of [(2097150, 18), 37748570, 18]-NRT-code), using
- digital (2, 10, 20)-net over F9, using
(123, 140, large)-Net over F9 — Digital
Digital (123, 140, large)-net over F9, using
- 93 times duplication [i] based on digital (120, 137, large)-net over F9, using
- t-expansion [i] based on digital (117, 137, large)-net over F9, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(9137, large, F9, 20) (dual of [large, large−137, 21]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523360 | 98−1, defining interval I = [0,19], and designed minimum distance d ≥ |I|+1 = 21 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(9137, large, F9, 20) (dual of [large, large−137, 21]-code), using
- t-expansion [i] based on digital (117, 137, large)-net over F9, using
(123, 140, large)-Net in Base 9 — Upper bound on s
There is no (123, 140, large)-net in base 9, because
- 15 times m-reduction [i] would yield (123, 125, large)-net in base 9, but