MinT - Best Known (89−35, 89, s)-Nets in Base 4

Best Known (89−35, 89, s)-Nets in Base 4

(89−35, 89, 130)-Net over F₄ — Constructive and digital

Digital (54, 89, 130)-net over F₄, using

7 times m-reduction [i] based on digital (54, 96, 130)-net over F₄, using
- trace code for nets [i] based on digital (6, 48, 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]

(89−35, 89, 159)-Net over F₄ — Digital

Digital (54, 89, 159)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(89−35, 89, 3115)-Net in Base 4 — Upper bound on s

There is no (54, 89, 3116)-net in base 4, because

1 times m-reduction [i] would yield (54, 88, 3116)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 96251 113597 896686 267728 312408 884902 511848 072196 551722 > 4⁸⁸ [i]