MinT - Best Known (183, 183+63, s)-Nets in Base 4

Best Known (183, 183+63, s)-Nets in Base 4

(183, 183+63, 548)-Net over F₄ — Constructive and digital

Digital (183, 246, 548)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (5, 36, 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 (147, 210, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 70, 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]

(183, 183+63, 648)-Net in Base 4 — Constructive

(183, 246, 648)-net in base 4, using

3 times m-reduction [i] based on (183, 249, 648)-net in base 4, using
- trace code for nets [i] based on (17, 83, 216)-net in base 64, using
  - 1 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]

(183, 183+63, 1984)-Net over F₄ — Digital

Digital (183, 246, 1984)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(183, 183+63, 237181)-Net in Base 4 — Upper bound on s

There is no (183, 246, 237182)-net in base 4, because

1 times m-reduction [i] would yield (183, 245, 237182)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3196 748845 112604 023931 529530 876507 969678 805722 445838 606217 231332 670399 650959 306620 687163 923805 650966 320676 723287 465569 503455 859006 973486 223904 035680 > 4²⁴⁵ [i]