MinT - Best Known (51, 51+35, s)-Nets in Base 4

Best Known (51, 51+35, s)-Nets in Base 4

(51, 51+35, 130)-Net over F₄ — Constructive and digital

Digital (51, 86, 130)-net over F₄, using

4 times m-reduction [i] based on digital (51, 90, 130)-net over F₄, using
- trace code for nets [i] based on digital (6, 45, 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]

(51, 51+35, 138)-Net over F₄ — Digital

Digital (51, 86, 138)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(51, 51+35, 2436)-Net in Base 4 — Upper bound on s

There is no (51, 86, 2437)-net in base 4, because

1 times m-reduction [i] would yield (51, 85, 2437)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 1505 692023 572465 665347 556064 319736 630672 720398 503680 > 4⁸⁵ [i]