Best Known (101, 134, s)-Nets in Base 3
(101, 134, 328)-Net over F3 — Constructive and digital
Digital (101, 134, 328)-net over F3, using
- 32 times duplication [i] based on digital (99, 132, 328)-net over F3, using
- trace code for nets [i] based on digital (0, 33, 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, 33, 82)-net over F81, using
(101, 134, 665)-Net over F3 — Digital
Digital (101, 134, 665)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3134, 665, F3, 33) (dual of [665, 531, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(3134, 743, F3, 33) (dual of [743, 609, 34]-code), using
- construction X applied to C([0,16]) ⊂ C([0,15]) [i] based on
- linear OA(3133, 730, F3, 33) (dual of [730, 597, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 730 | 312−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- linear OA(3121, 730, F3, 31) (dual of [730, 609, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 730 | 312−1, defining interval I = [0,15], and minimum distance d ≥ |{−15,−14,…,15}|+1 = 32 (BCH-bound) [i]
- linear OA(31, 13, F3, 1) (dual of [13, 12, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,16]) ⊂ C([0,15]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3134, 743, F3, 33) (dual of [743, 609, 34]-code), using
(101, 134, 31431)-Net in Base 3 — Upper bound on s
There is no (101, 134, 31432)-net in base 3, because
- 1 times m-reduction [i] would yield (101, 133, 31432)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 2865 745655 766023 625871 800971 862507 048393 981008 802965 632689 910785 > 3133 [i]