MinT - Best Known (68, 68+178, s)-Nets in Base 4

Best Known (68, 68+178, s)-Nets in Base 4

Digital (68, 246, 66)-net over F₄, using

t-expansion [i] based on digital (49, 246, 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, 246, 99)-net over F₄, using

t-expansion [i] based on digital (61, 246, 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, 246, 379)-net over F₄, because

2 times m-reduction [i] would yield digital (68, 244, 379)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²⁴⁴, 379, F₄, 176) (dual of [379, 135, 177]-code), but
  - residual code [i] would yield OA(4⁶⁸, 202, S₄, 44), but
    - the linear programming bound shows that MÂ â‰¥ 2027 088631 021403 337954 762092 477564 421959 685162 636703 100335 360507 890568 637631 560712 794323 981191 765389 475840 / 21729 825107 879568 901112 749356 633788 098469 459498 248484 364910 480751 > 4⁶⁸ [i]

There is no (68, 246, 452)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 13083 412167 990638 458523 348387 784305 169993 464776 084598 245514 498275 825523 439841 534952 372637 551406 130447 449991 625353 103903 344142 544549 854026 432509 530128 > 4²⁴⁶ [i]