MinT - Best Known (244−63, 244, s)-Nets in Base 4

Best Known (244−63, 244, s)-Nets in Base 4

(244−63, 244, 545)-Net over F₄ — Constructive and digital

Digital (181, 244, 545)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (3, 34, 14)-net over F₄, using
  - net from sequence [i] based on digital (3, 13)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 3 and N(F) ≥ 14, 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]

(244−63, 244, 648)-Net in Base 4 — Constructive

(181, 244, 648)-net in base 4, using

2 times m-reduction [i] based on (181, 246, 648)-net in base 4, using
- trace code for nets [i] based on (17, 82, 216)-net in base 64, using
  - 2 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]

(244−63, 244, 1898)-Net over F₄ — Digital

Digital (181, 244, 1898)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(244−63, 244, 216887)-Net in Base 4 — Upper bound on s

There is no (181, 244, 216888)-net in base 4, because

1 times m-reduction [i] would yield (181, 243, 216888)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 199 807024 753046 429823 939186 769394 679541 756785 571950 825886 073302 405394 867055 491953 431202 963389 820207 310417 826486 040611 791106 724800 577381 130254 434220 > 4²⁴³ [i]