MinT - Best Known (259−114, 259, s)-Nets in Base 4

Best Known (259−114, 259, s)-Nets in Base 4

(259−114, 259, 137)-Net over F₄ — Constructive and digital

Digital (145, 259, 137)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (15, 72, 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, 187, 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]

(259−114, 259, 270)-Net over F₄ — Digital

Digital (145, 259, 270)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(259−114, 259, 3957)-Net in Base 4 — Upper bound on s

There is no (145, 259, 3958)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 861422 400775 136679 784725 333745 567484 913317 920642 922686 906893 250001 656948 111873 785737 233156 764438 263617 207279 268088 578948 056069 854419 750897 001086 238549 886840 > 4²⁵⁹ [i]