MinT - Best Known (156, 156+53, s)-Nets in Base 4

Best Known (156, 156+53, s)-Nets in Base 4

(156, 156+53, 545)-Net over F₄ — Constructive and digital

Digital (156, 209, 545)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (3, 29, 14)-net over F₄, using
  - net from sequence [i] based on digital (3, 13)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 3 and N(F) ≥ 14, using
      - an explicitly constructive algebraic function field [i]
2. digital (127, 180, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 60, 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]

(156, 156+53, 648)-Net in Base 4 — Constructive

(156, 209, 648)-net in base 4, using

t-expansion [i] based on (155, 209, 648)-net in base 4, using
- 1 times m-reduction [i] based on (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]

(156, 156+53, 1798)-Net over F₄ — Digital

Digital (156, 209, 1798)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(156, 156+53, 230468)-Net in Base 4 — Upper bound on s

There is no (156, 209, 230469)-net in base 4, because

1 times m-reduction [i] would yield (156, 208, 230469)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 169237 269519 308456 718145 978635 202948 836151 346277 233850 329842 115391 916816 837862 394466 162373 901211 744132 943091 806210 284547 453104 > 4²⁰⁸ [i]