MinT - Best Known (191−48, 191, s)-Nets in Base 4

Best Known (191−48, 191, s)-Nets in Base 4

(191−48, 191, 541)-Net over F₄ — Constructive and digital

Digital (143, 191, 541)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (2, 26, 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 (117, 165, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 55, 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]

(191−48, 191, 648)-Net in Base 4 — Constructive

(143, 191, 648)-net in base 4, using

4² times duplication [i] based on (141, 189, 648)-net in base 4, using
- t-expansion [i] based on (140, 189, 648)-net in base 4, using
  - trace code for nets [i] based on (14, 63, 216)-net in base 64, using
    - base change [i] based on digital (5, 54, 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]

(191−48, 191, 1736)-Net over F₄ — Digital

Digital (143, 191, 1736)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(191−48, 191, 202112)-Net in Base 4 — Upper bound on s

There is no (143, 191, 202113)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 9 850810 037324 238669 231432 079675 844859 394280 939637 739811 444902 318778 471268 935868 716303 098218 902131 185576 917448 995056 > 4¹⁹¹ [i]