MinT - Best Known (65, 65+37, s)-Nets in Base 16

Best Known (65, 65+37, s)-Nets in Base 16

(65, 65+37, 583)-Net over F₁₆ — Constructive and digital

Digital (65, 102, 583)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (6, 24, 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]
2. digital (41, 78, 518)-net over F₁₆, using
  - trace code for nets [i] based on digital (2, 39, 259)-net over F₂₅₆, using
    - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]

(65, 65+37, 2474)-Net over F₁₆ — Digital

Digital (65, 102, 2474)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(65, 65+37, 2874010)-Net in Base 16 — Upper bound on s

There is no (65, 102, 2874011)-net in base 16, because

1 times m-reduction [i] would yield (65, 101, 2874011)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 41 316135 687371 576933 660227 741451 404672 805112 864980 990690 588658 933620 709826 282673 151347 356978 617206 792954 458439 192386 117771 > 16¹⁰¹ [i]