MinT - Best Known (87−31, 87, s)-Nets in Base 4

Best Known (87−31, 87, s)-Nets in Base 4

(87−31, 87, 130)-Net over F₄ — Constructive and digital

Digital (56, 87, 130)-net over F₄, using

13 times m-reduction [i] based on digital (56, 100, 130)-net over F₄, using
- trace code for nets [i] based on digital (6, 50, 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]

(87−31, 87, 219)-Net over F₄ — Digital

Digital (56, 87, 219)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(87−31, 87, 6047)-Net in Base 4 — Upper bound on s

There is no (56, 87, 6048)-net in base 4, because

1 times m-reduction [i] would yield (56, 86, 6048)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 5989 611419 490830 681284 884422 024949 476127 036159 299821 > 4⁸⁶ [i]