MinT - Best Known (82, 239, s)-Nets in Base 3

Best Known (82, 239, s)-Nets in Base 3

(82, 239, 57)-Net over F₃ — Constructive and digital

Digital (82, 239, 57)-net over F₃, using

net from sequence [i] based on digital (82, 56)-sequence over F₃, using
- base reduction for sequences [i] based on digital (13, 56)-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]

(82, 239, 84)-Net over F₃ — Digital

Digital (82, 239, 84)-net over F₃, using

t-expansion [i] based on digital (71, 239, 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]

(82, 239, 266)-Net over F₃ — Upper bound on s (digital)

There is no digital (82, 239, 267)-net over F₃, because

1 times m-reduction [i] would yield digital (82, 238, 267)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3²³⁸, 267, F₃, 156) (dual of [267, 29, 157]-code), but
  - residual code [i] would yield OA(3⁸², 110, S₃, 52), but
    - the linear programming bound shows that MÂ â‰¥ 36 916814 627242 513115 313210 304023 547918 724700 246946 892256 / 26125 454924 406409 > 3⁸² [i]

(82, 239, 354)-Net in Base 3 — Upper bound on s

There is no (82, 239, 355)-net in base 3, because

1 times m-reduction [i] would yield (82, 238, 355)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 410055 316762 625721 708931 484866 445478 091411 622182 106048 423654 781843 395731 365412 038343 535218 036776 279512 371881 657741 > 3²³⁸ [i]