MinT - Best Known (86, 86+159, s)-Nets in Base 3

Best Known (86, 86+159, s)-Nets in Base 3

(86, 86+159, 61)-Net over F₃ — Constructive and digital

Digital (86, 245, 61)-net over F₃, using

net from sequence [i] based on digital (86, 60)-sequence over F₃, using
- base reduction for sequences [i] based on digital (13, 60)-sequence over F₉, using
  - s-reduction based on digital (13, 63)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 13 and N(F) ≥ 64, using
      - a function field by GarcÃa and Quoos [i]

(86, 86+159, 84)-Net over F₃ — Digital

Digital (86, 245, 84)-net over F₃, using

t-expansion [i] based on digital (71, 245, 84)-net over F₃, using
- net from sequence [i] based on digital (71, 83)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 71 and N(F) ≥ 84, using
    - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(86, 86+159, 292)-Net over F₃ — Upper bound on s (digital)

There is no digital (86, 245, 293)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3²⁴⁵, 293, F₃, 159) (dual of [293, 48, 160]-code), but
- residual code [i] would yield OA(3⁸⁶, 133, S₃, 53), but
  - the linear programming bound shows that MÂ â‰¥ 146601 713206 672215 854529 961683 786203 805963 714360 771638 656247 204712 306825 / 1 191506 890413 541201 033837 596832 > 3⁸⁶ [i]

(86, 86+159, 376)-Net in Base 3 — Upper bound on s

There is no (86, 245, 377)-net in base 3, because

1 times m-reduction [i] would yield (86, 244, 377)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 285 560480 034805 230801 586931 453952 698941 264707 935185 765269 974296 944793 358230 115374 298304 739036 667953 607632 428511 582539 > 3²⁴⁴ [i]