MinT - Best Known (58, 58+176, s)-Nets in Base 4

Best Known (58, 58+176, s)-Nets in Base 4

Digital (58, 234, 66)-net over F₄, using

t-expansion [i] based on digital (49, 234, 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]

Digital (58, 234, 91)-net over F₄, using

t-expansion [i] based on digital (50, 234, 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]

There is no digital (58, 234, 241)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4²³⁴, 241, F₄, 176) (dual of [241, 7, 177]-code), but
- residual code [i] would yield linear OA(4⁵⁸, 64, F₄, 44) (dual of [64, 6, 45]-code), but
  - residual code [i] would yield linear OA(4¹⁴, 19, F₄, 11) (dual of [19, 5, 12]-code), but
    - 1 times truncation [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]

There is no (58, 234, 378)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 823 963997 235546 239884 231121 731413 628673 907953 976671 467654 867073 260581 317763 539439 371755 594667 446383 922178 441017 978488 456292 454363 244175 107505 > 4²³⁴ [i]