MinT - Best Known (89−37, 89, s)-Nets in Base 4

Best Known (89−37, 89, s)-Nets in Base 4

(89−37, 89, 130)-Net over F₄ — Constructive and digital

Digital (52, 89, 130)-net over F₄, using

3 times m-reduction [i] based on digital (52, 92, 130)-net over F₄, using
- trace code for nets [i] based on digital (6, 46, 65)-net over F₁₆, using
  - net from sequence [i] based on digital (6, 64)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 6 and N(F) ≥ 65, using
      - the Hermitian function field over F₁₆ [i]

(89−37, 89, 133)-Net over F₄ — Digital

Digital (52, 89, 133)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(89−37, 89, 2195)-Net in Base 4 — Upper bound on s

There is no (52, 89, 2196)-net in base 4, because

1 times m-reduction [i] would yield (52, 88, 2196)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 96059 655026 245645 034646 130070 230983 432376 252167 535819 > 4⁸⁸ [i]