MinT - Best Known (67, 67+139, s)-Nets in Base 4

Best Known (67, 67+139, s)-Nets in Base 4

(67, 67+139, 66)-Net over F₄ — Constructive and digital

Digital (67, 206, 66)-net over F₄, using

t-expansion [i] based on digital (49, 206, 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]

(67, 67+139, 99)-Net over F₄ — Digital

Digital (67, 206, 99)-net over F₄, using

t-expansion [i] based on digital (61, 206, 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]

(67, 67+139, 488)-Net in Base 4 — Upper bound on s

There is no (67, 206, 489)-net in base 4, because

1 times m-reduction [i] would yield (67, 205, 489)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2725 363607 715112 337955 941847 389214 045945 136124 695529 206191 995285 968530 565438 005803 181172 579602 602023 753975 360092 768666 356320 > 4²⁰⁵ [i]