MinT - Best Known (211−81, 211, s)-Nets in Base 4

Best Known (211−81, 211, s)-Nets in Base 4

(211−81, 211, 137)-Net over F₄ — Constructive and digital

Digital (130, 211, 137)-net over F₄, using

3 times m-reduction [i] based on digital (130, 214, 137)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 57, 33)-net over F₄, using
    - net from sequence [i] based on digital (15, 32)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 15 and N(F) ≥ 33, using
        an explicitly constructive algebraic function field [i]
  2. digital (73, 157, 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]

(211−81, 211, 351)-Net over F₄ — Digital

Digital (130, 211, 351)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(211−81, 211, 7578)-Net in Base 4 — Upper bound on s

There is no (130, 211, 7579)-net in base 4, because

1 times m-reduction [i] would yield (130, 210, 7579)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2 708022 776396 468079 262496 108729 913737 284393 629966 535310 894092 703138 535067 527055 315989 576822 274053 447669 191028 164448 194637 361732 > 4²¹⁰ [i]