MinT - Best Known (66, 257, s)-Nets in Base 4

Best Known (66, 257, s)-Nets in Base 4

Digital (66, 257, 66)-net over F₄, using

t-expansion [i] based on digital (49, 257, 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, 257, 99)-net over F₄, using

t-expansion [i] based on digital (61, 257, 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, 257, 296)-net over F₄, because

3 times m-reduction [i] would yield digital (66, 254, 296)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²⁵⁴, 296, F₄, 188) (dual of [296, 42, 189]-code), but
  - residual code [i] would yield OA(4⁶⁶, 107, S₄, 47), but
    - the linear programming bound shows that MÂ â‰¥ 877667 617526 699604 115459 206661 960900 469804 406919 637481 696619 379668 118193 820628 156416 / 152 206215 688292 487856 759376 046017 491747 653675 > 4⁶⁶ [i]

There is no (66, 257, 431)-net in base 4, because

1 times m-reduction [i] would yield (66, 256, 431)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 14233 596721 796052 675498 562508 867830 750179 262872 398138 032907 965911 646598 960234 382883 753563 183978 393174 286076 725686 200663 073664 845330 026656 571092 971999 813040 > 4²⁵⁶ [i]