MinT - Best Known (236−170, 236, s)-Nets in Base 4

Best Known (236−170, 236, s)-Nets in Base 4

Digital (66, 236, 66)-net over F₄, using

t-expansion [i] based on digital (49, 236, 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 (66, 236, 99)-net over F₄, using

t-expansion [i] based on digital (61, 236, 99)-net over F₄, using
- net from sequence [i] based on digital (61, 98)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 61 and N(F) ≥ 99, using
    - Drinfeld modules of rank 1 [i]

There is no digital (66, 236, 383)-net over F₄, because

2 times m-reduction [i] would yield digital (66, 234, 383)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²³⁴, 383, F₄, 168) (dual of [383, 149, 169]-code), but
  - residual code [i] would yield OA(4⁶⁶, 214, S₄, 42), but
    - the linear programming bound shows that MÂ â‰¥ 113 026714 592523 165071 661888 079481 443471 353704 422593 024055 302731 631761 970405 969474 549718 427811 971072 / 20291 149710 796120 930866 391420 785164 794765 888342 457078 864235 > 4⁶⁶ [i]

There is no (66, 236, 441)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 12424 019490 504078 216852 525724 808003 976641 365603 790813 315948 421994 523278 763102 543124 136912 616594 541510 609924 769210 872265 761878 324298 314855 533120 > 4²³⁶ [i]