MinT - Best Known (252−93, 252, s)-Nets in Base 4

Best Known (252−93, 252, s)-Nets in Base 4

(252−93, 252, 160)-Net over F₄ — Constructive and digital

Digital (159, 252, 160)-net over F₄, using

t-expansion [i] based on digital (157, 252, 160)-net over F₄, using
- 7 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]

(252−93, 252, 196)-Net in Base 4 — Constructive

(159, 252, 196)-net in base 4, using

4² times duplication [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]

(252−93, 252, 471)-Net over F₄ — Digital

Digital (159, 252, 471)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(252−93, 252, 11530)-Net in Base 4 — Upper bound on s

There is no (159, 252, 11531)-net in base 4, because

1 times m-reduction [i] would yield (159, 251, 11531)-net in base 4, but
- 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]