Best Known (112, 112+9, s)-Nets in Base 3
(112, 112+9, 4194304)-Net over F3 — Constructive and digital
Digital (112, 121, 4194304)-net over F3, using
- generalized (u, u+v)-construction [i] based on
- digital (0, 3, 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 (24, 28, 2097150)-net over F3, using
- s-reduction based on digital (24, 28, 2391485)-net over F3, using
- digital (81, 90, 2097150)-net over F3, using
- net defined by OOA [i] based on linear OOA(390, 2097150, F3, 9, 9) (dual of [(2097150, 9), 18874260, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(390, 8388601, F3, 9) (dual of [8388601, 8388511, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(390, large, F3, 9) (dual of [large, large−90, 10]-code), using
- the primitive narrow-sense BCH-code C(I) with length 14348906 = 315−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- discarding factors / shortening the dual code based on linear OA(390, large, F3, 9) (dual of [large, large−90, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(390, 8388601, F3, 9) (dual of [8388601, 8388511, 10]-code), using
- net defined by OOA [i] based on linear OOA(390, 2097150, F3, 9, 9) (dual of [(2097150, 9), 18874260, 10]-NRT-code), using
- digital (0, 3, 4)-net over F3, using
(112, 112+9, large)-Net over F3 — Digital
Digital (112, 121, large)-net over F3, using
- 38 times duplication [i] based on digital (104, 113, large)-net over F3, using
- t-expansion [i] based on digital (103, 113, large)-net over F3, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3113, large, F3, 10) (dual of [large, large−113, 11]-code), using
- 22 times code embedding in larger space [i] based on linear OA(391, large, F3, 10) (dual of [large, large−91, 11]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 14348906 = 315−1, defining interval I = [0,9], and designed minimum distance d ≥ |I|+1 = 11 [i]
- 22 times code embedding in larger space [i] based on linear OA(391, large, F3, 10) (dual of [large, large−91, 11]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3113, large, F3, 10) (dual of [large, large−113, 11]-code), using
- t-expansion [i] based on digital (103, 113, large)-net over F3, using
(112, 112+9, large)-Net in Base 3 — Upper bound on s
There is no (112, 121, large)-net in base 3, because
- 7 times m-reduction [i] would yield (112, 114, large)-net in base 3, but