MinT - Best Known (246−64, 246, s)-Nets in Base 4

Best Known (246−64, 246, s)-Nets in Base 4

(246−64, 246, 540)-Net over F₄ — Constructive and digital

Digital (182, 246, 540)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (1, 33, 9)-net over F₄, using
  - net from sequence [i] based on digital (1, 8)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 1 and N(F) ≥ 9, using
      - the Hermitian function field over F₄ [i]
2. digital (149, 213, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 71, 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]

(246−64, 246, 648)-Net in Base 4 — Constructive

(182, 246, 648)-net in base 4, using

t-expansion [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]

(246−64, 246, 1849)-Net over F₄ — Digital

Digital (182, 246, 1849)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(246−64, 246, 181147)-Net in Base 4 — Upper bound on s

There is no (182, 246, 181148)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 12787 599014 200057 305573 134500 284307 316420 129489 053393 192381 379805 996611 370297 574801 939388 244016 570515 667864 386527 207426 067263 788277 034898 930288 741852 > 4²⁴⁶ [i]