Best Known (159−39, 159, s)-Nets in Base 3
(159−39, 159, 328)-Net over F3 — Constructive and digital
Digital (120, 159, 328)-net over F3, using
- 1 times m-reduction [i] based on digital (120, 160, 328)-net over F3, using
- trace code for nets [i] based on digital (0, 40, 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, 40, 82)-net over F81, using
(159−39, 159, 766)-Net over F3 — Digital
Digital (120, 159, 766)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3159, 766, F3, 39) (dual of [766, 607, 40]-code), using
- discarding factors / shortening the dual code based on linear OA(3159, 771, F3, 39) (dual of [771, 612, 40]-code), using
- 37 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 0, 0, 0, 1, 6 times 0, 1, 9 times 0, 1, 13 times 0) [i] based on linear OA(3153, 728, F3, 39) (dual of [728, 575, 40]-code), using
- the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,39], and designed minimum distance d ≥ |I|+1 = 40 [i]
- 37 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 0, 0, 0, 1, 6 times 0, 1, 9 times 0, 1, 13 times 0) [i] based on linear OA(3153, 728, F3, 39) (dual of [728, 575, 40]-code), using
- discarding factors / shortening the dual code based on linear OA(3159, 771, F3, 39) (dual of [771, 612, 40]-code), using
(159−39, 159, 36779)-Net in Base 3 — Upper bound on s
There is no (120, 159, 36780)-net in base 3, because
- 1 times m-reduction [i] would yield (120, 158, 36780)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 2427 988522 767242 329508 386881 139923 579477 452303 492844 852274 178313 286620 589137 > 3158 [i]