MinT - Best Known (241−61, 241, s)-Nets in Base 4

Best Known (241−61, 241, s)-Nets in Base 4

(241−61, 241, 552)-Net over F₄ — Constructive and digital

Digital (180, 241, 552)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (7, 37, 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 (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]

(241−61, 241, 648)-Net in Base 4 — Constructive

(180, 241, 648)-net in base 4, using

t-expansion [i] based on (179, 241, 648)-net in base 4, using
- 2 times m-reduction [i] based on (179, 243, 648)-net in base 4, using
  - trace code for nets [i] based on (17, 81, 216)-net in base 64, using
    - 3 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]

(241−61, 241, 2055)-Net over F₄ — Digital

Digital (180, 241, 2055)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(241−61, 241, 263091)-Net in Base 4 — Upper bound on s

There is no (180, 241, 263092)-net in base 4, because

1 times m-reduction [i] would yield (180, 240, 263092)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3 121907 476972 280129 647525 914678 636545 828369 982052 389668 471008 452882 883426 249194 275946 051814 737163 480280 011576 060406 797670 417869 274861 033933 304424 > 4²⁴⁰ [i]