MinT - Best Known (138−71, 138, s)-Nets in Base 4

Best Known (138−71, 138, s)-Nets in Base 4

(138−71, 138, 66)-Net over F₄ — Constructive and digital

Digital (67, 138, 66)-net over F₄, using

t-expansion [i] based on digital (49, 138, 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]

(138−71, 138, 99)-Net over F₄ — Digital

Digital (67, 138, 99)-net over F₄, using

t-expansion [i] based on digital (61, 138, 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]

(138−71, 138, 1025)-Net in Base 4 — Upper bound on s

There is no (67, 138, 1026)-net in base 4, because

1 times m-reduction [i] would yield (67, 137, 1026)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 30888 212912 521627 232718 057813 113062 433564 261962 973094 449440 454941 911293 389223 898112 > 4¹³⁷ [i]