Best Known (190−29, 190, s)-Nets in Base 3
(190−29, 190, 1480)-Net over F3 — Constructive and digital
Digital (161, 190, 1480)-net over F3, using
- t-expansion [i] based on digital (160, 190, 1480)-net over F3, using
- 2 times m-reduction [i] based on digital (160, 192, 1480)-net over F3, using
- trace code for nets [i] based on digital (16, 48, 370)-net over F81, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 16 and N(F) ≥ 370, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- trace code for nets [i] based on digital (16, 48, 370)-net over F81, using
- 2 times m-reduction [i] based on digital (160, 192, 1480)-net over F3, using
(190−29, 190, 11920)-Net over F3 — Digital
Digital (161, 190, 11920)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3190, 11920, F3, 29) (dual of [11920, 11730, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(3190, 19703, F3, 29) (dual of [19703, 19513, 30]-code), using
- (u, u+v)-construction [i] based on
- linear OA(318, 20, F3, 14) (dual of [20, 2, 15]-code), using
- repeating each code word 5 times [i] based on linear OA(32, 4, F3, 2) (dual of [4, 2, 3]-code or 4-arc in PG(1,3)), using
- extended Reed–Solomon code RSe(2,3) [i]
- Hamming code H(2,3) [i]
- Simplex code S(2,3) [i]
- the Tetracode [i]
- repeating each code word 5 times [i] based on linear OA(32, 4, F3, 2) (dual of [4, 2, 3]-code or 4-arc in PG(1,3)), using
- linear OA(3172, 19683, F3, 29) (dual of [19683, 19511, 30]-code), using
- an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(318, 20, F3, 14) (dual of [20, 2, 15]-code), using
- (u, u+v)-construction [i] based on
- discarding factors / shortening the dual code based on linear OA(3190, 19703, F3, 29) (dual of [19703, 19513, 30]-code), using
(190−29, 190, 8347645)-Net in Base 3 — Upper bound on s
There is no (161, 190, 8347646)-net in base 3, because
- 1 times m-reduction [i] would yield (161, 189, 8347646)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 1 499400 841791 836005 421869 670306 196494 090926 206926 497945 854203 734549 324975 072614 912360 333621 > 3189 [i]