MinT - Best Known (237−101, 237, s)-Nets in Base 4

Best Known (237−101, 237, s)-Nets in Base 4

(237−101, 237, 134)-Net over F₄ — Constructive and digital

Digital (136, 237, 134)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (13, 63, 30)-net over F₄, using
  - net from sequence [i] based on digital (13, 29)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 13 and N(F) ≥ 30, using
      - F₄ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]
2. digital (73, 174, 104)-net over F₄, using
  - net from sequence [i] based on digital (73, 103)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 73 and N(F) ≥ 104, using
      - F₆ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(237−101, 237, 277)-Net over F₄ — Digital

Digital (136, 237, 277)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(237−101, 237, 4469)-Net in Base 4 — Upper bound on s

There is no (136, 237, 4470)-net in base 4, because

1 times m-reduction [i] would yield (136, 236, 4470)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 12210 763507 229743 310051 865280 532184 505084 960026 439418 966692 951816 660071 348842 890801 427357 337442 667860 559907 427074 846480 308123 047788 138140 331790 > 4²³⁶ [i]