MinT - Best Known (109, 109+54, s)-Nets in Base 4

Best Known (109, 109+54, s)-Nets in Base 4

(109, 109+54, 195)-Net over F₄ — Constructive and digital

Digital (109, 163, 195)-net over F₄, using

4¹ times duplication [i] based on digital (108, 162, 195)-net over F₄, using
- trace code for nets [i] based on digital (0, 54, 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, 109+54, 208)-Net in Base 4 — Constructive

(109, 163, 208)-net in base 4, using

3 times m-reduction [i] based on (109, 166, 208)-net in base 4, using
- trace code for nets [i] based on (26, 83, 104)-net in base 16, using
  - 2 times m-reduction [i] based on (26, 85, 104)-net in base 16, using
    - base change [i] based on digital (9, 68, 104)-net over F₃₂, using
      - net from sequence [i] based on digital (9, 103)-sequence over F₃₂, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 9 and N(F) ≥ 104, using
        a function field by SÃ©mirat [i]

(109, 109+54, 472)-Net over F₄ — Digital

Digital (109, 163, 472)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(109, 109+54, 15679)-Net in Base 4 — Upper bound on s

There is no (109, 163, 15680)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 136 793827 940771 348880 239538 458712 632609 978026 068851 605649 014068 211739 620985 697684 658399 377884 248285 > 4¹⁶³ [i]