MinT - Best Known (30, 51, s)-Nets in Base 9

Best Known (30, 51, s)-Nets in Base 9

Digital (30, 51, 300)-net over F₉, using

1 times m-reduction [i] based on digital (30, 52, 300)-net over F₉, using
- trace code for nets [i] based on digital (4, 26, 150)-net over F₈₁, using
  - net from sequence [i] based on digital (4, 149)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 4 and N(F) ≥ 150, using
      - a function field by SÃ©mirat [i]

Digital (30, 51, 312)-net over F₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(9⁵¹, 312, F₉, 21) (dual of [312, 261, 22]-code), using
- 3 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (1, 0, 0) [i] based on linear OA(9⁵⁰, 308, F₉, 21) (dual of [308, 258, 22]-code), using
  - trace code [i] based on linear OA(81²⁵, 154, F₈₁, 21) (dual of [154, 129, 22]-code), using
    - extended algebraic-geometric code AG_e(F,132P) [i] based on function field F/F₈₁ with g(F) = 4 and N(F) ≥ 154, using
      - a curve from the manYPoints database [i]

There is no (30, 51, 33422)-net in base 9, because

1 times m-reduction [i] would yield (30, 50, 33422)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 515475 764503 408892 643242 135307 923481 761218 279841 > 9⁵⁰ [i]