MinT - Best Known (174−58, 174, s)-Nets in Base 4

Best Known (174−58, 174, s)-Nets in Base 4

(174−58, 174, 195)-Net over F₄ — Constructive and digital

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]

(174−58, 174, 240)-Net in Base 4 — Constructive

(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]

(174−58, 174, 488)-Net over F₄ — Digital

Digital (116, 174, 488)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(174−58, 174, 15911)-Net in Base 4 — Upper bound on s

There is no (116, 174, 15912)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 573 724978 753021 250696 189609 537315 411026 099031 350123 608752 248669 483432 959471 106409 308739 816950 214941 932748 > 4¹⁷⁴ [i]