MinT - Best Known (155, 155+55, s)-Nets in Base 4

Best Known (155, 155+55, s)-Nets in Base 4

(155, 155+55, 531)-Net over F₄ — Constructive and digital

Digital (155, 210, 531)-net over F₄, using

12 times m-reduction [i] based on digital (155, 222, 531)-net over F₄, using
- trace code for nets [i] based on digital (7, 74, 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]

(155, 155+55, 648)-Net in Base 4 — Constructive

(155, 210, 648)-net in base 4, using

trace code for nets [i] based on (15, 70, 216)-net in base 64, using
- base change [i] based on digital (5, 60, 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]

(155, 155+55, 1561)-Net over F₄ — Digital

Digital (155, 210, 1561)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(155, 155+55, 166578)-Net in Base 4 — Upper bound on s

There is no (155, 210, 166579)-net in base 4, because

1 times m-reduction [i] would yield (155, 209, 166579)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 676996 326775 770607 698207 899510 576496 816245 084034 169956 313573 747871 073573 681782 516717 237774 294598 028263 466980 966317 178292 439920 > 4²⁰⁹ [i]