MinT - Best Known (239−61, 239, s)-Nets in Base 4

Best Known (239−61, 239, s)-Nets in Base 4

(239−61, 239, 548)-Net over F₄ — Constructive and digital

Digital (178, 239, 548)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (5, 35, 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 (143, 204, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 68, 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]

(239−61, 239, 648)-Net in Base 4 — Constructive

(178, 239, 648)-net in base 4, using

t-expansion [i] based on (177, 239, 648)-net in base 4, using
- 1 times m-reduction [i] based on (177, 240, 648)-net in base 4, using
  - trace code for nets [i] based on (17, 80, 216)-net in base 64, using
    - 4 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]

(239−61, 239, 1964)-Net over F₄ — Digital

Digital (178, 239, 1964)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(239−61, 239, 239864)-Net in Base 4 — Upper bound on s

There is no (178, 239, 239865)-net in base 4, because

1 times m-reduction [i] would yield (178, 238, 239865)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 195125 797848 978797 182751 849759 161940 053488 858997 847411 417147 881340 632820 505253 771917 817947 248660 338843 376860 569775 219766 817199 170350 661860 054320 > 4²³⁸ [i]