MinT - Best Known (58, 58+147, s)-Nets in Base 4

Best Known (58, 58+147, s)-Nets in Base 4

Digital (58, 205, 66)-net over F₄, using

t-expansion [i] based on digital (49, 205, 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, 205, 91)-net over F₄, using

t-expansion [i] based on digital (50, 205, 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, 205, 358)-net over F₄, because

3 times m-reduction [i] would yield digital (58, 202, 358)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²⁰², 358, F₄, 144) (dual of [358, 156, 145]-code), but
  - residual code [i] would yield OA(4⁵⁸, 213, S₄, 36), but
    - the linear programming bound shows that MÂ â‰¥ 7 866958 687285 876635 384332 274577 437607 350342 946368 588838 836611 539357 984012 369920 / 88 856238 935032 449112 554626 779870 493844 853867 > 4⁵⁸ [i]

There is no (58, 205, 392)-net in base 4, because

1 times m-reduction [i] would yield (58, 204, 392)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 667 997153 036560 355197 722561 245934 903292 750366 927041 055197 541674 089560 416077 200835 241971 290846 754523 021035 372361 310264 660432 > 4²⁰⁴ [i]