MinT - Best Known (68, 205, s)-Nets in Base 4

Best Known (68, 205, s)-Nets in Base 4

(68, 205, 66)-Net over F₄ — Constructive and digital

Digital (68, 205, 66)-net over F₄, using

t-expansion [i] based on digital (49, 205, 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, 205, 99)-Net over F₄ — Digital

Digital (68, 205, 99)-net over F₄, using

t-expansion [i] based on digital (61, 205, 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, 205, 503)-Net in Base 4 — Upper bound on s

There is no (68, 205, 504)-net in base 4, because

1 times m-reduction [i] would yield (68, 204, 504)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 671 528610 807398 585753 725070 842852 558828 127511 029354 752960 532747 120658 311198 997307 356150 660526 783725 261894 904506 000508 055983 > 4²⁰⁴ [i]