MinT - Best Known (230−175, 230, s)-Nets in Base 4

Best Known (230−175, 230, s)-Nets in Base 4

(230−175, 230, 66)-Net over F₄ — Constructive and digital

Digital (55, 230, 66)-net over F₄, using

t-expansion [i] based on digital (49, 230, 66)-net over F₄, using
- net from sequence [i] based on digital (49, 65)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 49 and N(F) ≥ 66, using
    - T₆ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(230−175, 230, 91)-Net over F₄ — Digital

Digital (55, 230, 91)-net over F₄, using

t-expansion [i] based on digital (50, 230, 91)-net over F₄, using
- net from sequence [i] based on digital (50, 90)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 50 and N(F) ≥ 91, using
    - a curve from the manYPoints database [i]

(230−175, 230, 229)-Net over F₄ — Upper bound on s (digital)

There is no digital (55, 230, 230)-net over F₄, because

7 times m-reduction [i] would yield digital (55, 223, 230)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²²³, 230, F₄, 168) (dual of [230, 7, 169]-code), but
  - residual code [i] would yield linear OA(4⁵⁵, 61, F₄, 42) (dual of [61, 6, 43]-code), but
    - 2 times truncation [i] would yield linear OA(4⁵³, 59, F₄, 40) (dual of [59, 6, 41]-code), but
      - residual code [i] would yield linear OA(4¹³, 18, F₄, 10) (dual of [18, 5, 11]-code), but
        â€œLizâ€ bound on codes from Brouwerâ€™s database [i]

(230−175, 230, 357)-Net in Base 4 — Upper bound on s

There is no (55, 230, 358)-net in base 4, because

1 times m-reduction [i] would yield (55, 229, 358)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 832075 533437 964791 521156 593986 686345 081376 809882 089402 952289 960762 315241 335938 817603 789585 149659 355885 584404 062445 412617 360386 207725 444400 > 4²²⁹ [i]