MinT - Best Known (82, 229, s)-Nets in Base 3

Best Known (82, 229, s)-Nets in Base 3

(82, 229, 57)-Net over F₃ — Constructive and digital

Digital (82, 229, 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, 229, 84)-Net over F₃ — Digital

Digital (82, 229, 84)-net over F₃, using

t-expansion [i] based on digital (71, 229, 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, 229, 354)-Net over F₃ — Upper bound on s (digital)

There is no digital (82, 229, 355)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3²²⁹, 355, F₃, 147) (dual of [355, 126, 148]-code), but
- residual code [i] would yield linear OA(3⁸², 207, F₃, 49) (dual of [207, 125, 50]-code), but
  - 1 times truncation [i] would yield linear OA(3⁸¹, 206, F₃, 48) (dual of [206, 125, 49]-code), but
    - the Johnson bound shows that NÂ â‰¤ 390067 621362 525166 675859 888729 736992 372634 079787 564789 713745 < 3¹²⁵ [i]

(82, 229, 365)-Net in Base 3 — Upper bound on s

There is no (82, 229, 366)-net in base 3, because

1 times m-reduction [i] would yield (82, 228, 366)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 7 116237 145420 157732 724459 455277 847574 767784 530774 936787 153866 931648 845010 419524 411828 600009 341464 696041 376461 > 3²²⁸ [i]