MinT - Best Known (137−67, 137, s)-Nets in Base 4

Best Known (137−67, 137, s)-Nets in Base 4

(137−67, 137, 71)-Net over F₄ — Constructive and digital

Digital (70, 137, 71)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (4, 37, 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, 100, 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]

(137−67, 137, 111)-Net over F₄ — Digital

Digital (70, 137, 111)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(137−67, 137, 1301)-Net in Base 4 — Upper bound on s

There is no (70, 137, 1302)-net in base 4, because

1 times m-reduction [i] would yield (70, 136, 1302)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 7592 622150 560872 026497 571202 687359 770123 937262 765526 840754 381683 931375 362123 844760 > 4¹³⁶ [i]