Best Known (159−11, 159, s)-Nets in Base 3
(159−11, 159, 5033167)-Net over F3 — Constructive and digital
Digital (148, 159, 5033167)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (1, 6, 7)-net over F3, using
- net from sequence [i] based on digital (1, 6)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 1 and N(F) ≥ 7, using
- net from sequence [i] based on digital (1, 6)-sequence over F3, using
- digital (142, 153, 5033160)-net over F3, using
- trace code for nets [i] based on digital (40, 51, 1677720)-net over F27, using
- net defined by OOA [i] based on linear OOA(2751, 1677720, F27, 11, 11) (dual of [(1677720, 11), 18454869, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(2751, 8388601, F27, 11) (dual of [8388601, 8388550, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(2751, large, F27, 11) (dual of [large, large−51, 12]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 14348908 | 2710−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(2751, large, F27, 11) (dual of [large, large−51, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(2751, 8388601, F27, 11) (dual of [8388601, 8388550, 12]-code), using
- net defined by OOA [i] based on linear OOA(2751, 1677720, F27, 11, 11) (dual of [(1677720, 11), 18454869, 12]-NRT-code), using
- trace code for nets [i] based on digital (40, 51, 1677720)-net over F27, using
- digital (1, 6, 7)-net over F3, using
(159−11, 159, large)-Net over F3 — Digital
Digital (148, 159, large)-net over F3, using
- 37 times duplication [i] based on digital (141, 152, large)-net over F3, using
- t-expansion [i] based on digital (139, 152, large)-net over F3, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3152, large, F3, 13) (dual of [large, large−152, 14]-code), using
- 31 times code embedding in larger space [i] based on linear OA(3121, large, F3, 13) (dual of [large, large−121, 14]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 14348908 | 330−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- 31 times code embedding in larger space [i] based on linear OA(3121, large, F3, 13) (dual of [large, large−121, 14]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3152, large, F3, 13) (dual of [large, large−152, 14]-code), using
- t-expansion [i] based on digital (139, 152, large)-net over F3, using
(159−11, 159, large)-Net in Base 3 — Upper bound on s
There is no (148, 159, large)-net in base 3, because
- 9 times m-reduction [i] would yield (148, 150, large)-net in base 3, but