MinT - Best Known (251−90, 251, s)-Nets in Base 4

Best Known (251−90, 251, s)-Nets in Base 4

(251−90, 251, 200)-Net over F₄ — Constructive and digital

Digital (161, 251, 200)-net over F₄, using

net from sequence [i] based on digital (161, 199)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 161 and N(F) ≥ 200, using
  - F₇ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(251−90, 251, 208)-Net in Base 4 — Constructive

(161, 251, 208)-net in base 4, using

4¹ times duplication [i] based on (160, 250, 208)-net in base 4, using
- t-expansion [i] based on (159, 250, 208)-net in base 4, using
  - trace code for nets [i] based on (34, 125, 104)-net in base 16, using
    - base change [i] based on digital (9, 100, 104)-net over F₃₂, using
      - net from sequence [i] based on digital (9, 103)-sequence over F₃₂, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 9 and N(F) ≥ 104, using
        a function field by SÃ©mirat [i]

(251−90, 251, 519)-Net over F₄ — Digital

Digital (161, 251, 519)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(251−90, 251, 13366)-Net in Base 4 — Upper bound on s

There is no (161, 251, 13367)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 13 126124 350290 037992 615724 983913 530278 857975 821459 471796 821743 987698 092266 586254 512975 830021 643527 285932 265181 927502 966951 015934 610622 681088 095810 344564 > 4²⁵¹ [i]