MinT - Best Known (175−58, 175, s)-Nets in Base 4

Best Known (175−58, 175, s)-Nets in Base 4

(175−58, 175, 195)-Net over F₄ — Constructive and digital

Digital (117, 175, 195)-net over F₄, using

4¹ times duplication [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]

(175−58, 175, 240)-Net in Base 4 — Constructive

(117, 175, 240)-net in base 4, using

1 times m-reduction [i] based on (117, 176, 240)-net in base 4, using
- trace code for nets [i] based on (29, 88, 120)-net in base 16, using
  - 2 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]

(175−58, 175, 501)-Net over F₄ — Digital

Digital (117, 175, 501)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(175−58, 175, 16691)-Net in Base 4 — Upper bound on s

There is no (117, 175, 16692)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2293 798289 628668 045763 242669 637754 296450 042912 182841 528130 506598 240176 799911 194951 475580 653468 108176 701720 > 4¹⁷⁵ [i]