MinT - Best Known (227−57, 227, s)-Nets in Base 4

Best Known (227−57, 227, s)-Nets in Base 4

(227−57, 227, 552)-Net over F₄ — Constructive and digital

Digital (170, 227, 552)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (7, 35, 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 (135, 192, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 64, 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]

(227−57, 227, 648)-Net in Base 4 — Constructive

(170, 227, 648)-net in base 4, using

4 times m-reduction [i] based on (170, 231, 648)-net in base 4, using
- trace code for nets [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]

(227−57, 227, 2023)-Net over F₄ — Digital

Digital (170, 227, 2023)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(227−57, 227, 272478)-Net in Base 4 — Upper bound on s

There is no (170, 227, 272479)-net in base 4, because

1 times m-reduction [i] would yield (170, 226, 272479)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 11630 497008 393037 429875 929878 199406 273559 480302 307449 262610 906019 682102 669162 645004 758464 011336 866716 313053 522375 147623 996087 327363 490607 > 4²²⁶ [i]