MinT - Best Known (72, 113, s)-Nets in Base 16

Best Known (72, 113, s)-Nets in Base 16

(72, 113, 583)-Net over F₁₆ — Constructive and digital

Digital (72, 113, 583)-net over F₁₆, using

16¹ times duplication [i] based on digital (71, 112, 583)-net over F₁₆, using
- (u,Â u+v)-construction [i] based on
  1. digital (6, 26, 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 (45, 86, 518)-net over F₁₆, using
    - trace code for nets [i] based on digital (2, 43, 259)-net over F₂₅₆, using
      - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
        Niederreiter sequence [i]

(72, 113, 2671)-Net over F₁₆ — Digital

Digital (72, 113, 2671)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(72, 113, 3063975)-Net in Base 16 — Upper bound on s

There is no (72, 113, 3063976)-net in base 16, because

1 times m-reduction [i] would yield (72, 112, 3063976)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 726 839394 978603 997513 723142 023635 230310 735921 594284 595773 325256 836601 755355 835281 470225 157990 499562 946923 482405 470692 295073 387765 467426 > 16¹¹² [i]