MinT - Best Known (145, 145+109, s)-Nets in Base 4

Best Known (145, 145+109, s)-Nets in Base 4

(145, 145+109, 137)-Net over F₄ — Constructive and digital

Digital (145, 254, 137)-net over F₄, using

5 times m-reduction [i] based on 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]

(145, 145+109, 288)-Net over F₄ — Digital

Digital (145, 254, 288)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(145, 145+109, 4581)-Net in Base 4 — Upper bound on s

There is no (145, 254, 4582)-net in base 4, because

1 times m-reduction [i] would yield (145, 253, 4582)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 210 883538 862597 297296 633190 007449 159516 279618 054415 783606 300742 247219 379612 874183 666088 964057 222732 907740 203144 356394 331993 286330 551555 157899 606526 494720 > 4²⁵³ [i]