MinT - Best Known (235−59, 235, s)-Nets in Base 4

Best Known (235−59, 235, s)-Nets in Base 4

(235−59, 235, 552)-Net over F₄ — Constructive and digital

Digital (176, 235, 552)-net over F₄, using

4¹ times duplication [i] based on digital (175, 234, 552)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (7, 36, 21)-net over F₄, using
    - net from sequence [i] based on digital (7, 20)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 7 and N(F) ≥ 21, using
        an explicitly constructive algebraic function field [i]
  2. digital (139, 198, 531)-net over F₄, using
    - trace code for nets [i] based on digital (7, 66, 177)-net over F₆₄, using
      - net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
        a function field by GarcÃa and Quoos [i]

(235−59, 235, 648)-Net in Base 4 — Constructive

(176, 235, 648)-net in base 4, using

t-expansion [i] based on (175, 235, 648)-net in base 4, using
- 2 times m-reduction [i] based on (175, 237, 648)-net in base 4, using
  - trace code for nets [i] based on (17, 79, 216)-net in base 64, using
    - 5 times m-reduction [i] based on (17, 84, 216)-net in base 64, using
      - base change [i] based on digital (5, 72, 216)-net over F₁₂₈, using
        net from sequence [i] based on digital (5, 215)-sequence over F₁₂₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 5 and N(F) ≥ 216, using
        a function field by SÃ©mirat [i]

(235−59, 235, 2087)-Net over F₄ — Digital

Digital (176, 235, 2087)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(235−59, 235, 280521)-Net in Base 4 — Upper bound on s

There is no (176, 235, 280522)-net in base 4, because

1 times m-reduction [i] would yield (176, 234, 280522)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 762 221009 786056 576572 762636 308086 361469 546800 011840 865793 453884 720371 801168 995684 570270 549964 258271 245660 803247 438066 042220 992084 578103 932976 > 4²³⁴ [i]