MinT - Best Known (68, 68+135, s)-Nets in Base 4

Best Known (68, 68+135, s)-Nets in Base 4

(68, 68+135, 66)-Net over F₄ — Constructive and digital

Digital (68, 203, 66)-net over F₄, using

t-expansion [i] based on digital (49, 203, 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]

(68, 68+135, 99)-Net over F₄ — Digital

Digital (68, 203, 99)-net over F₄, using

t-expansion [i] based on digital (61, 203, 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]

(68, 68+135, 508)-Net in Base 4 — Upper bound on s

There is no (68, 203, 509)-net in base 4, because

1 times m-reduction [i] would yield (68, 202, 509)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 45 282832 714928 616544 617919 263868 577074 744864 897061 459170 534970 221320 388116 283271 798737 103673 310135 945716 112883 121163 219900 > 4²⁰² [i]