MinT - Best Known (230−97, 230, s)-Nets in Base 4

Best Known (230−97, 230, s)-Nets in Base 4

(230−97, 230, 132)-Net over F₄ — Constructive and digital

Digital (133, 230, 132)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (12, 60, 28)-net over F₄, using
  - net from sequence [i] based on digital (12, 27)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 12 and N(F) ≥ 28, using
      - an explicitly constructive algebraic function field [i]
2. digital (73, 170, 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]

(230−97, 230, 279)-Net over F₄ — Digital

Digital (133, 230, 279)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(230−97, 230, 4616)-Net in Base 4 — Upper bound on s

There is no (133, 230, 4617)-net in base 4, because

1 times m-reduction [i] would yield (133, 229, 4617)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 748515 761426 804558 635627 017100 805151 486436 627379 162841 421994 658614 596311 162430 457682 644661 294761 775287 461079 074212 420544 019408 674872 491872 > 4²²⁹ [i]