MinT - Best Known (191, 191+69, s)-Nets in Base 4

Best Known (191, 191+69, s)-Nets in Base 4

(191, 191+69, 531)-Net over F₄ — Constructive and digital

Digital (191, 260, 531)-net over F₄, using

4² times duplication [i] based on digital (189, 258, 531)-net over F₄, using
- t-expansion [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]

(191, 191+69, 576)-Net in Base 4 — Constructive

(191, 260, 576)-net in base 4, using

4² times duplication [i] based on (189, 258, 576)-net in base 4, using
- t-expansion [i] based on (188, 258, 576)-net in base 4, using
  - trace code for nets [i] based on (16, 86, 192)-net in base 64, using
    - 5 times m-reduction [i] based on (16, 91, 192)-net in base 64, using
      - base change [i] based on digital (3, 78, 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]

(191, 191+69, 1776)-Net over F₄ — Digital

Digital (191, 260, 1776)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(191, 191+69, 174006)-Net in Base 4 — Upper bound on s

There is no (191, 260, 174007)-net in base 4, because

1 times m-reduction [i] would yield (191, 259, 174007)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 858199 166578 287988 041332 598435 587222 054046 362899 655372 515985 336384 236071 924010 072100 045212 426547 563892 173440 548239 496471 695504 759502 567398 895441 779749 909517 > 4²⁵⁹ [i]