MinT - Best Known (68, 68+78, s)-Nets in Base 4

Best Known (68, 68+78, s)-Nets in Base 4

(68, 68+78, 66)-Net over F₄ — Constructive and digital

Digital (68, 146, 66)-net over F₄, using

t-expansion [i] based on digital (49, 146, 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+78, 99)-Net over F₄ — Digital

Digital (68, 146, 99)-net over F₄, using

t-expansion [i] based on digital (61, 146, 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+78, 889)-Net in Base 4 — Upper bound on s

There is no (68, 146, 890)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 8221 717843 550488 664819 955641 888919 669110 758940 554384 779049 001315 018317 544169 026337 445012 > 4¹⁴⁶ [i]