MinT - Best Known (250−65, 250, s)-Nets in Base 4

Best Known (250−65, 250, s)-Nets in Base 4

(250−65, 250, 541)-Net over F₄ — Constructive and digital

Digital (185, 250, 541)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (2, 34, 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 (151, 216, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 72, 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]

(250−65, 250, 648)-Net in Base 4 — Constructive

(185, 250, 648)-net in base 4, using

2 times m-reduction [i] based on (185, 252, 648)-net in base 4, using
- trace code for nets [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]

(250−65, 250, 1881)-Net over F₄ — Digital

Digital (185, 250, 1881)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(250−65, 250, 206292)-Net in Base 4 — Upper bound on s

There is no (185, 250, 206293)-net in base 4, because

1 times m-reduction [i] would yield (185, 249, 206293)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 818417 143366 179168 123991 973903 234444 660600 788981 018927 749049 950358 534683 192731 315551 136249 172303 266545 417371 158562 309646 566414 905695 974251 905666 895038 > 4²⁴⁹ [i]