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

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

(140−69, 140, 71)-Net over F₄ — Constructive and digital

Digital (71, 140, 71)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (4, 38, 15)-net over F₄, using
  - net from sequence [i] based on digital (4, 14)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 4 and N(F) ≥ 15, using
      - an explicitly constructive algebraic function field [i]
2. digital (33, 102, 56)-net over F₄, using
  - net from sequence [i] based on digital (33, 55)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 33 and N(F) ≥ 56, using
      - F₅ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(140−69, 140, 111)-Net over F₄ — Digital

Digital (71, 140, 111)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(140−69, 140, 1277)-Net in Base 4 — Upper bound on s

There is no (71, 140, 1278)-net in base 4, because

1 times m-reduction [i] would yield (71, 139, 1278)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 489321 075124 859194 126503 359398 122017 862284 012770 267527 875290 618279 497181 660663 887838 > 4¹³⁹ [i]