MinT - Best Known (85, 85+42, s)-Nets in Base 4

Best Known (85, 85+42, s)-Nets in Base 4

(85, 85+42, 195)-Net over F₄ — Constructive and digital

Digital (85, 127, 195)-net over F₄, using

4¹ times duplication [i] based on digital (84, 126, 195)-net over F₄, using
- trace code for nets [i] based on digital (0, 42, 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]

(85, 85+42, 196)-Net in Base 4 — Constructive

(85, 127, 196)-net in base 4, using

3 times m-reduction [i] based on (85, 130, 196)-net in base 4, using
- trace code for nets [i] based on (20, 65, 98)-net in base 16, using
  - base change [i] based on digital (7, 52, 98)-net over F₃₂, using
    - net from sequence [i] based on digital (7, 97)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 7 and N(F) ≥ 98, using
        a function field by SÃ©mirat [i]

(85, 85+42, 388)-Net over F₄ — Digital

Digital (85, 127, 388)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(85, 85+42, 12642)-Net in Base 4 — Upper bound on s

There is no (85, 127, 12643)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 28993 717766 620174 786762 101056 001424 106423 873156 890459 917699 860188 839151 707760 > 4¹²⁷ [i]