MinT - Best Known (249−67, 249, s)-Nets in Base 4

Best Known (249−67, 249, s)-Nets in Base 4

(249−67, 249, 531)-Net over F₄ — Constructive and digital

Digital (182, 249, 531)-net over F₄, using

t-expansion [i] based on digital (179, 249, 531)-net over F₄, using
- 9 times m-reduction [i] based on digital (179, 258, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 86, 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]

(249−67, 249, 576)-Net in Base 4 — Constructive

(182, 249, 576)-net in base 4, using

t-expansion [i] based on (181, 249, 576)-net in base 4, using
- trace code for nets [i] based on (15, 83, 192)-net in base 64, using
  - 1 times m-reduction [i] based on (15, 84, 192)-net in base 64, using
    - base change [i] based on digital (3, 72, 192)-net over F₁₂₈, using
      - net from sequence [i] based on digital (3, 191)-sequence over F₁₂₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 3 and N(F) ≥ 192, using
        a function field by SÃ©mirat [i]

(249−67, 249, 1606)-Net over F₄ — Digital

Digital (182, 249, 1606)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(249−67, 249, 146799)-Net in Base 4 — Upper bound on s

There is no (182, 249, 146800)-net in base 4, because

1 times m-reduction [i] would yield (182, 248, 146800)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 204602 561775 559163 115953 042630 034377 180189 157450 648658 773050 707361 010878 042485 008000 887422 543809 130632 019240 441822 863094 897074 588303 512351 311897 600627 > 4²⁴⁸ [i]