MinT - Best Known (82, 228, s)-Nets in Base 3

Best Known (82, 228, s)-Nets in Base 3

(82, 228, 57)-Net over F₃ — Constructive and digital

Digital (82, 228, 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, 228, 84)-Net over F₃ — Digital

Digital (82, 228, 84)-net over F₃, using

t-expansion [i] based on digital (71, 228, 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, 228, 359)-Net over F₃ — Upper bound on s (digital)

There is no digital (82, 228, 360)-net over F₃, because

2 times m-reduction [i] would yield digital (82, 226, 360)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3²²⁶, 360, F₃, 144) (dual of [360, 134, 145]-code), but
  - residual code [i] would yield linear OA(3⁸², 215, F₃, 48) (dual of [215, 133, 49]-code), but
    - the Johnson bound shows that NÂ â‰¤ 2633 391490 584373 459265 026071 777086 821369 449862 527703 301746 510338 < 3¹³³ [i]

(82, 228, 365)-Net in Base 3 — Upper bound on s

There is no (82, 228, 366)-net in base 3, because

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]