MinT - Best Known (65, 65+170, s)-Nets in Base 4

Best Known (65, 65+170, s)-Nets in Base 4

Digital (65, 235, 66)-net over F₄, using

t-expansion [i] based on digital (49, 235, 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 (65, 235, 99)-net over F₄, using

t-expansion [i] based on digital (61, 235, 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 (65, 235, 365)-net over F₄, because

2 times m-reduction [i] would yield digital (65, 233, 365)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²³³, 365, F₄, 168) (dual of [365, 132, 169]-code), but
  - residual code [i] would yield OA(4⁶⁵, 196, S₄, 42), but
    - the linear programming bound shows that MÂ â‰¥ 396900 403316 059479 106808 484395 402977 413951 648726 549399 691598 760260 216224 172313 340280 832000 / 276 682488 299206 750393 648129 881706 051242 679804 397123 > 4⁶⁵ [i]

There is no (65, 235, 433)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3193 642765 951666 519168 533250 523426 880313 709417 767827 843135 122663 250733 232190 740304 569233 225358 460714 954956 650106 588764 548041 020624 911127 146880 > 4²³⁵ [i]