MinT - Best Known (112−53, 112, s)-Nets in Base 4

Best Known (112−53, 112, s)-Nets in Base 4

(112−53, 112, 76)-Net over F₄ — Constructive and digital

Digital (59, 112, 76)-net over F₄, using

trace code for nets [i] based on digital (3, 56, 38)-net over F₁₆, using
- net from sequence [i] based on digital (3, 37)-sequence over F₁₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 3 and N(F) ≥ 38, using
    - a function field by SÃ©mirat [i]

(112−53, 112, 105)-Net over F₄ — Digital

Digital (59, 112, 105)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(112−53, 112, 1286)-Net in Base 4 — Upper bound on s

There is no (59, 112, 1287)-net in base 4, because

1 times m-reduction [i] would yield (59, 111, 1287)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 6 786476 986225 038028 516665 380850 965695 932699 001649 410591 429313 478304 > 4¹¹¹ [i]