MinT - Best Known (254−108, 254, s)-Nets in Base 4

Best Known (254−108, 254, s)-Nets in Base 4

(254−108, 254, 137)-Net over F₄ — Constructive and digital

Digital (146, 254, 137)-net over F₄, using

t-expansion [i] based on 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]

(254−108, 254, 297)-Net over F₄ — Digital

Digital (146, 254, 297)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(254−108, 254, 4701)-Net in Base 4 — Upper bound on s

There is no (146, 254, 4702)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 840 646502 084691 982985 830512 855593 528562 962902 667150 862077 351688 474711 369352 388594 249868 508424 849832 771641 592844 128077 693736 556920 024426 545397 322645 032520 > 4²⁵⁴ [i]