MinT - Best Known (164−59, 164, s)-Nets in Base 4

Best Known (164−59, 164, s)-Nets in Base 4

(164−59, 164, 147)-Net over F₄ — Constructive and digital

Digital (105, 164, 147)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (5, 34, 17)-net over F₄, using
  - net from sequence [i] based on digital (5, 16)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 5 and N(F) ≥ 17, using
      - an explicitly constructive algebraic function field [i]
2. digital (71, 130, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 65, 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]

(164−59, 164, 152)-Net in Base 4 — Constructive

(105, 164, 152)-net in base 4, using

2 times m-reduction [i] based on (105, 166, 152)-net in base 4, using
- trace code for nets [i] based on (22, 83, 76)-net in base 16, using
  - 2 times m-reduction [i] based on (22, 85, 76)-net in base 16, using
    - base change [i] based on digital (5, 68, 76)-net over F₃₂, using
      - net from sequence [i] based on digital (5, 75)-sequence over F₃₂, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 5 and N(F) ≥ 76, using
        a function field by SÃ©mirat [i]

(164−59, 164, 353)-Net over F₄ — Digital

Digital (105, 164, 353)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(164−59, 164, 9395)-Net in Base 4 — Upper bound on s

There is no (105, 164, 9396)-net in base 4, because

1 times m-reduction [i] would yield (105, 163, 9396)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 137 081084 944890 071293 573660 362566 418533 223946 537912 174561 491695 411128 755841 859366 432030 554530 778744 > 4¹⁶³ [i]