Best Known (121, 121+33, s)-Nets in Base 3
(121, 121+33, 640)-Net over F3 — Constructive and digital
Digital (121, 154, 640)-net over F3, using
- 32 times duplication [i] based on digital (119, 152, 640)-net over F3, using
- trace code for nets [i] based on digital (5, 38, 160)-net over F81, using
- net from sequence [i] based on digital (5, 159)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 5 and N(F) ≥ 160, using
- net from sequence [i] based on digital (5, 159)-sequence over F81, using
- trace code for nets [i] based on digital (5, 38, 160)-net over F81, using
(121, 121+33, 1377)-Net over F3 — Digital
Digital (121, 154, 1377)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3154, 1377, F3, 33) (dual of [1377, 1223, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(3154, 2186, F3, 33) (dual of [2186, 2032, 34]-code), using
- 1 times truncation [i] based on linear OA(3155, 2187, F3, 34) (dual of [2187, 2032, 35]-code), using
- an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- 1 times truncation [i] based on linear OA(3155, 2187, F3, 34) (dual of [2187, 2032, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(3154, 2186, F3, 33) (dual of [2186, 2032, 34]-code), using
(121, 121+33, 124144)-Net in Base 3 — Upper bound on s
There is no (121, 154, 124145)-net in base 3, because
- 1 times m-reduction [i] would yield (121, 153, 124145)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 9 990691 229498 266689 934136 477874 050173 780139 562766 459859 821953 285243 145025 > 3153 [i]