MinT - Best Known (164−55, 164, s)-Nets in Base 4

Best Known (164−55, 164, s)-Nets in Base 4

(164−55, 164, 163)-Net over F₄ — Constructive and digital

Digital (109, 164, 163)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (15, 42, 33)-net over F₄, using
  - net from sequence [i] based on digital (15, 32)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 15 and N(F) ≥ 33, using
      - an explicitly constructive algebraic function field [i]
2. digital (67, 122, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 61, 65)-net over F₁₆, using
    - net from sequence [i] based on digital (6, 64)-sequence over F₁₆, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 6 and N(F) ≥ 65, using
        the Hermitian function field over F₁₆ [i]

(164−55, 164, 208)-Net in Base 4 — Constructive

(109, 164, 208)-net in base 4, using

2 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]

(164−55, 164, 454)-Net over F₄ — Digital

Digital (109, 164, 454)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(164−55, 164, 15679)-Net in Base 4 — Upper bound on s

There is no (109, 164, 15680)-net in base 4, because

1 times m-reduction [i] would yield (109, 163, 15680)-net in base 4, but
- 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]