MinT - Best Known (68, 68+177, s)-Nets in Base 4

Best Known (68, 68+177, s)-Nets in Base 4

Digital (68, 245, 66)-net over F₄, using

t-expansion [i] based on digital (49, 245, 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, 245, 99)-net over F₄, using

t-expansion [i] based on digital (61, 245, 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, 245, 379)-net over F₄, because

1 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, 245, 454)-net in base 4, because

1 times m-reduction [i] would yield (68, 244, 454)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 918 590267 259728 598797 605847 290484 604358 273072 845549 643461 937323 112920 340634 357137 006302 321945 169182 787668 405922 644550 206824 350327 116811 302425 494384 > 4²⁴⁴ [i]