Best Known (145−13, 145, s)-Nets in Base 3
(145−13, 145, 1398428)-Net over F3 — Constructive and digital
Digital (132, 145, 1398428)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (18, 24, 328)-net over F3, using
- trace code for nets [i] based on digital (0, 6, 82)-net over F81, using
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 0 and N(F) ≥ 82, using
- the rational function field F81(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- trace code for nets [i] based on digital (0, 6, 82)-net over F81, using
- digital (108, 121, 1398100)-net over F3, using
- net defined by OOA [i] based on linear OOA(3121, 1398100, F3, 13, 13) (dual of [(1398100, 13), 18175179, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(3121, 8388601, F3, 13) (dual of [8388601, 8388480, 14]-code), using
- discarding factors / shortening the dual code 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]
- discarding factors / shortening the dual code based on linear OA(3121, large, F3, 13) (dual of [large, large−121, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(3121, 8388601, F3, 13) (dual of [8388601, 8388480, 14]-code), using
- net defined by OOA [i] based on linear OOA(3121, 1398100, F3, 13, 13) (dual of [(1398100, 13), 18175179, 14]-NRT-code), using
- digital (18, 24, 328)-net over F3, using
(145−13, 145, 4324482)-Net over F3 — Digital
Digital (132, 145, 4324482)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3145, 4324482, F3, 13) (dual of [4324482, 4324337, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(3145, large, F3, 13) (dual of [large, large−145, 14]-code), using
- 24 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]
- 24 times code embedding in larger space [i] based on linear OA(3121, large, F3, 13) (dual of [large, large−121, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(3145, large, F3, 13) (dual of [large, large−145, 14]-code), using
(145−13, 145, large)-Net in Base 3 — Upper bound on s
There is no (132, 145, large)-net in base 3, because
- 11 times m-reduction [i] would yield (132, 134, large)-net in base 3, but