MinT - Best Known (81, 81+159, s)-Nets in Base 3

Best Known (81, 81+159, s)-Nets in Base 3

(81, 81+159, 56)-Net over F₃ — Constructive and digital

Digital (81, 240, 56)-net over F₃, using

net from sequence [i] based on digital (81, 55)-sequence over F₃, using
- base reduction for sequences [i] based on digital (13, 55)-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]

(81, 81+159, 84)-Net over F₃ — Digital

Digital (81, 240, 84)-net over F₃, using

t-expansion [i] based on digital (71, 240, 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]

(81, 81+159, 255)-Net over F₃ — Upper bound on s (digital)

There is no digital (81, 240, 256)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3²⁴⁰, 256, F₃, 159) (dual of [256, 16, 160]-code), but
- residual code [i] would yield OA(3⁸¹, 96, S₃, 53), but
  - the linear programming bound shows that MÂ â‰¥ 35667 308007 855330 590206 634210 852728 315487 161419 / 70 241195 > 3⁸¹ [i]

(81, 81+159, 346)-Net in Base 3 — Upper bound on s

There is no (81, 240, 347)-net in base 3, because

1 times m-reduction [i] would yield (81, 239, 347)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1 153829 933840 444879 287755 393489 317228 879428 419350 718979 194356 882770 231745 906552 575459 837856 294328 582995 132851 069395 > 3²³⁹ [i]