MinT - Best Known (83, 83+162, s)-Nets in Base 3

Best Known (83, 83+162, s)-Nets in Base 3

(83, 83+162, 58)-Net over F₃ — Constructive and digital

Digital (83, 245, 58)-net over F₃, using

net from sequence [i] based on digital (83, 57)-sequence over F₃, using
- base reduction for sequences [i] based on digital (13, 57)-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]

(83, 83+162, 84)-Net over F₃ — Digital

Digital (83, 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]

(83, 83+162, 261)-Net over F₃ — Upper bound on s (digital)

There is no digital (83, 245, 262)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3²⁴⁵, 262, F₃, 162) (dual of [262, 17, 163]-code), but
- residual code [i] would yield OA(3⁸³, 99, S₃, 54), but
  - the linear programming bound shows that MÂ â‰¥ 212325 169720 869780 184660 026483 230346 001188 329823 / 51 112831 > 3⁸³ [i]

(83, 83+162, 354)-Net in Base 3 — Upper bound on s

There is no (83, 245, 355)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 821 273667 653553 214919 116619 507716 225945 788984 341016 564826 650038 770362 736772 209121 364575 825473 328756 234026 547646 952103 > 3²⁴⁵ [i]