MinT - Best Known (83, 83+166, s)-Nets in Base 3

Best Known (83, 83+166, s)-Nets in Base 3

(83, 83+166, 58)-Net over F₃ — Constructive and digital

Digital (83, 249, 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+166, 84)-Net over F₃ — Digital

Digital (83, 249, 84)-net over F₃, using

t-expansion [i] based on digital (71, 249, 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+166, 260)-Net over F₃ — Upper bound on s (digital)

There is no digital (83, 249, 261)-net over F₃, because

1 times m-reduction [i] would yield digital (83, 248, 261)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3²⁴⁸, 261, F₃, 165) (dual of [261, 13, 166]-code), but
  - residual code [i] would yield OA(3⁸³, 95, S₃, 55), but
    - the linear programming bound shows that MÂ â‰¥ 648 778625 227853 290324 593879 307770 065609 747709 / 118144 > 3⁸³ [i]

(83, 83+166, 351)-Net in Base 3 — Upper bound on s

There is no (83, 249, 352)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 69475 798687 697457 315145 950780 513459 774541 930051 118985 332435 726441 408416 466356 428552 969094 632327 660431 471994 964630 397313 > 3²⁴⁹ [i]