MinT - Best Known (89, 89+59, s)-Nets in Base 4

Best Known (89, 89+59, s)-Nets in Base 4

(89, 89+59, 130)-Net over F₄ — Constructive and digital

Digital (89, 148, 130)-net over F₄, using

18 times m-reduction [i] based on digital (89, 166, 130)-net over F₄, using
- trace code for nets [i] based on digital (6, 83, 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, 89+59, 228)-Net over F₄ — Digital

Digital (89, 148, 228)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(89, 89+59, 4359)-Net in Base 4 — Upper bound on s

There is no (89, 148, 4360)-net in base 4, because

1 times m-reduction [i] would yield (89, 147, 4360)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 31851 136460 940064 401880 381081 461550 029664 173063 544049 166563 637724 009203 989709 642683 032096 > 4¹⁴⁷ [i]