MinT - Best Known (227−148, 227, s)-Nets in Base 3

Best Known (227−148, 227, s)-Nets in Base 3

(227−148, 227, 54)-Net over F₃ — Constructive and digital

Digital (79, 227, 54)-net over F₃, using

net from sequence [i] based on digital (79, 53)-sequence over F₃, using
- base reduction for sequences [i] based on digital (13, 53)-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]

(227−148, 227, 84)-Net over F₃ — Digital

Digital (79, 227, 84)-net over F₃, using

t-expansion [i] based on digital (71, 227, 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]

(227−148, 227, 269)-Net over F₃ — Upper bound on s (digital)

There is no digital (79, 227, 270)-net over F₃, because

1 times m-reduction [i] would yield digital (79, 226, 270)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3²²⁶, 270, F₃, 147) (dual of [270, 44, 148]-code), but
  - residual code [i] would yield OA(3⁷⁹, 122, S₃, 49), but
    - the linear programming bound shows that MÂ â‰¥ 2 696619 084610 776501 344031 056269 137248 595352 418845 365791 957429 212653 / 47840 556342 766214 122108 158565 > 3⁷⁹ [i]

(227−148, 227, 344)-Net in Base 3 — Upper bound on s

There is no (79, 227, 345)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 2 390786 114429 456430 705425 797529 327317 208819 819312 975655 086097 300794 150685 392869 819231 299732 783695 903761 008921 > 3²²⁷ [i]