MinT - Best Known (68, 68+171, s)-Nets in Base 4

Best Known (68, 68+171, s)-Nets in Base 4

Digital (68, 239, 66)-net over F₄, using

t-expansion [i] based on digital (49, 239, 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 (68, 239, 99)-net over F₄, using

t-expansion [i] based on digital (61, 239, 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 (68, 239, 415)-net over F₄, because

3 times m-reduction [i] would yield digital (68, 236, 415)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²³⁶, 415, F₄, 168) (dual of [415, 179, 169]-code), but
  - residual code [i] would yield OA(4⁶⁸, 246, S₄, 42), but
    - the linear programming bound shows that MÂ â‰¥ 8 266893 710032 655425 717892 686988 538675 685493 888138 140590 751265 026314 680838 406117 785600 000000 / 94 513486 949920 285641 128495 398550 971508 431933 193771 > 4⁶⁸ [i]

There is no (68, 239, 458)-net in base 4, because

1 times m-reduction [i] would yield (68, 238, 458)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 207562 429594 450256 899237 037444 590987 047736 131473 983993 301704 980975 961248 502397 168101 819386 622341 704476 353104 621706 005616 002156 128136 360555 116720 > 4²³⁸ [i]