MinT - Best Known (173−57, 173, s)-Nets in Base 4

Best Known (173−57, 173, s)-Nets in Base 4

(173−57, 173, 195)-Net over F₄ — Constructive and digital

Digital (116, 173, 195)-net over F₄, using

1 times m-reduction [i] based on digital (116, 174, 195)-net over F₄, using
- trace code for nets [i] based on digital (0, 58, 65)-net over F₆₄, using
  - net from sequence [i] based on digital (0, 64)-sequence over F₆₄, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 0 and N(F) ≥ 65, using
      - the rational function field F₆₄(x) [i]
    - Niederreiter sequence [i]

(173−57, 173, 240)-Net in Base 4 — Constructive

(116, 173, 240)-net in base 4, using

1 times m-reduction [i] based on (116, 174, 240)-net in base 4, using
- trace code for nets [i] based on (29, 87, 120)-net in base 16, using
  - 3 times m-reduction [i] based on (29, 90, 120)-net in base 16, using
    - base change [i] based on digital (11, 72, 120)-net over F₃₂, using
      - net from sequence [i] based on digital (11, 119)-sequence over F₃₂, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 11 and N(F) ≥ 120, using
        a function field by SÃ©mirat [i]

(173−57, 173, 507)-Net over F₄ — Digital

Digital (116, 173, 507)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(173−57, 173, 18781)-Net in Base 4 — Upper bound on s

There is no (116, 173, 18782)-net in base 4, because

1 times m-reduction [i] would yield (116, 172, 18782)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 35 875098 078892 563406 978867 624481 163454 894001 882549 936905 115911 218549 993000 443765 525557 968844 632743 176561 > 4¹⁷² [i]