MinT - Best Known (243−63, 243, s)-Nets in Base 4

Best Known (243−63, 243, s)-Nets in Base 4

(243−63, 243, 541)-Net over F₄ — Constructive and digital

Digital (180, 243, 541)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (2, 33, 10)-net over F₄, using
  - net from sequence [i] based on digital (2, 9)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 2 and N(F) ≥ 10, 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]

(243−63, 243, 648)-Net in Base 4 — Constructive

(180, 243, 648)-net in base 4, using

t-expansion [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]

(243−63, 243, 1857)-Net over F₄ — Digital

Digital (180, 243, 1857)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(243−63, 243, 207401)-Net in Base 4 — Upper bound on s

There is no (180, 243, 207402)-net in base 4, because

1 times m-reduction [i] would yield (180, 242, 207402)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 49 955413 545448 766447 965215 605227 540984 454019 060595 457691 962292 993976 973796 797475 957629 902298 249492 337568 712973 080386 049866 199941 331458 986811 248496 > 4²⁴² [i]