MinT - Best Known (109−36, 109, s)-Nets in Base 4

Best Known (109−36, 109, s)-Nets in Base 4

(109−36, 109, 195)-Net over F₄ — Constructive and digital

Digital (73, 109, 195)-net over F₄, using

4¹ times duplication [i] based on digital (72, 108, 195)-net over F₄, using
- trace code for nets [i] based on digital (0, 36, 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]

(109−36, 109, 196)-Net in Base 4 — Constructive

(73, 109, 196)-net in base 4, using

1 times m-reduction [i] based on (73, 110, 196)-net in base 4, using
- trace code for nets [i] based on (18, 55, 98)-net in base 16, using
  - base change [i] based on digital (7, 44, 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]

(109−36, 109, 346)-Net over F₄ — Digital

Digital (73, 109, 346)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(109−36, 109, 11123)-Net in Base 4 — Upper bound on s

There is no (73, 109, 11124)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 421421 997085 201343 175300 788620 680526 310049 491359 522292 192371 352497 > 4¹⁰⁹ [i]