Best Known (83, 83+29, s)-Nets in Base 3
(83, 83+29, 252)-Net over F3 — Constructive and digital
Digital (83, 112, 252)-net over F3, using
- 31 times duplication [i] based on digital (82, 111, 252)-net over F3, using
- trace code for nets [i] based on digital (8, 37, 84)-net over F27, using
- net from sequence [i] based on digital (8, 83)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 8 and N(F) ≥ 84, using
- net from sequence [i] based on digital (8, 83)-sequence over F27, using
- trace code for nets [i] based on digital (8, 37, 84)-net over F27, using
(83, 83+29, 477)-Net over F3 — Digital
Digital (83, 112, 477)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3112, 477, F3, 29) (dual of [477, 365, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(3112, 728, F3, 29) (dual of [728, 616, 30]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [0,28], and designed minimum distance d ≥ |I|+1 = 30 [i]
- discarding factors / shortening the dual code based on linear OA(3112, 728, F3, 29) (dual of [728, 616, 30]-code), using
(83, 83+29, 18323)-Net in Base 3 — Upper bound on s
There is no (83, 112, 18324)-net in base 3, because
- 1 times m-reduction [i] would yield (83, 111, 18324)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 91363 343129 807054 026477 187703 481802 724822 793838 414393 > 3111 [i]