MinT - Best Known (191, 191+68, s)-Nets in Base 4

Best Known (191, 191+68, s)-Nets in Base 4

(191, 191+68, 536)-Net over F₄ — Constructive and digital

Digital (191, 259, 536)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (0, 34, 5)-net over F₄, using
  - net from sequence [i] based on digital (0, 4)-sequence over F₄, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 0 and N(F) ≥ 5, using
      - the rational function field F₄(x) [i]
    - Niederreiter sequence [i]
2. digital (157, 225, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 75, 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+68, 648)-Net in Base 4 — Constructive

(191, 259, 648)-net in base 4, using

4¹ times duplication [i] based on (190, 258, 648)-net in base 4, using
- trace code for nets [i] based on (18, 86, 216)-net in base 64, using
  - 5 times m-reduction [i] based on (18, 91, 216)-net in base 64, using
    - base change [i] based on digital (5, 78, 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, 191+68, 1860)-Net over F₄ — Digital

Digital (191, 259, 1860)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(191, 191+68, 174006)-Net in Base 4 — Upper bound on s

There is no (191, 259, 174007)-net in base 4, because

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]