MinT - Best Known (251−92, 251, s)-Nets in Base 4

Best Known (251−92, 251, s)-Nets in Base 4

(251−92, 251, 160)-Net over F₄ — Constructive and digital

Digital (159, 251, 160)-net over F₄, using

t-expansion [i] based on digital (157, 251, 160)-net over F₄, using
- 8 times m-reduction [i] based on digital (157, 259, 160)-net over F₄, using
  - (u,Â u+v)-construction [i] based on
    1. digital (33, 84, 56)-net over F₄, using
      - net from sequence [i] based on digital (33, 55)-sequence over F₄, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 33 and N(F) ≥ 56, using
        F₅ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]
    2. digital (73, 175, 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]

(251−92, 251, 196)-Net in Base 4 — Constructive

(159, 251, 196)-net in base 4, using

4¹ times duplication [i] based on (158, 250, 196)-net in base 4, using
- t-expansion [i] based on (157, 250, 196)-net in base 4, using
  - trace code for nets [i] based on (32, 125, 98)-net in base 16, using
    - base change [i] based on digital (7, 100, 98)-net over F₃₂, using
      - net from sequence [i] based on digital (7, 97)-sequence over F₃₂, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 7 and N(F) ≥ 98, using
        a function field by SÃ©mirat [i]

(251−92, 251, 480)-Net over F₄ — Digital

Digital (159, 251, 480)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(251−92, 251, 11530)-Net in Base 4 — Upper bound on s

There is no (159, 251, 11531)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 13 129443 423093 534525 519112 743325 033279 575857 637668 744146 739926 155931 647943 458475 731500 663482 814798 851254 123663 879965 162367 500432 897756 509922 856336 014272 > 4²⁵¹ [i]