MinT - Best Known (65, 65+41, s)-Nets in Base 4

Best Known (65, 65+41, s)-Nets in Base 4

(65, 65+41, 130)-Net over F₄ — Constructive and digital

Digital (65, 106, 130)-net over F₄, using

12 times m-reduction [i] based on digital (65, 118, 130)-net over F₄, using
- trace code for nets [i] based on digital (6, 59, 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]

(65, 65+41, 192)-Net over F₄ — Digital

Digital (65, 106, 192)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(65, 65+41, 3992)-Net in Base 4 — Upper bound on s

There is no (65, 106, 3993)-net in base 4, because

1 times m-reduction [i] would yield (65, 105, 3993)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 1649 408357 326773 131489 826777 406187 126710 746380 336917 036280 680594 > 4¹⁰⁵ [i]