MinT - Best Known (215−54, 215, s)-Nets in Base 4

Best Known (215−54, 215, s)-Nets in Base 4

(215−54, 215, 548)-Net over F₄ — Constructive and digital

Digital (161, 215, 548)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (5, 32, 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 (129, 183, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 61, 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]

(215−54, 215, 648)-Net in Base 4 — Constructive

(161, 215, 648)-net in base 4, using

t-expansion [i] based on (160, 215, 648)-net in base 4, using
- 1 times m-reduction [i] based on (160, 216, 648)-net in base 4, using
  - trace code for nets [i] based on (16, 72, 216)-net in base 64, using
    - 5 times m-reduction [i] based on (16, 77, 216)-net in base 64, using
      - base change [i] based on digital (5, 66, 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]

(215−54, 215, 1927)-Net over F₄ — Digital

Digital (161, 215, 1927)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(215−54, 215, 226686)-Net in Base 4 — Upper bound on s

There is no (161, 215, 226687)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2772 985501 637620 301060 797221 825457 047846 454399 833123 853182 118487 920834 082334 872422 481495 855551 355420 579778 884595 803603 883666 823600 > 4²¹⁵ [i]