MinT - Best Known (69, 140, s)-Nets in Base 4

Best Known (69, 140, s)-Nets in Base 4

(69, 140, 66)-Net over F₄ — Constructive and digital

Digital (69, 140, 66)-net over F₄, using

t-expansion [i] based on digital (49, 140, 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]

(69, 140, 101)-Net over F₄ — Digital

Digital (69, 140, 101)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(69, 140, 1112)-Net in Base 4 — Upper bound on s

There is no (69, 140, 1113)-net in base 4, because

1 times m-reduction [i] would yield (69, 139, 1113)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 495742 182636 915624 192616 839627 597718 892751 557518 918489 418880 024011 796729 057030 290128 > 4¹³⁹ [i]