MinT - Best Known (35, 82, s)-Nets in Base 16

Best Known (35, 82, s)-Nets in Base 16

(35, 82, 130)-Net over F₁₆ — Constructive and digital

Digital (35, 82, 130)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (6, 29, 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 (6, 53, 65)-net over F₁₆, using
  - net from sequence [i] based on digital (6, 64)-sequence over F₁₆ (see above)

(35, 82, 177)-Net in Base 16 — Constructive

(35, 82, 177)-net in base 16, using

2 times m-reduction [i] based on (35, 84, 177)-net in base 16, using
- base change [i] based on digital (7, 56, 177)-net over F₆₄, using
  - net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
      - a function field by GarcÃa and Quoos [i]

(35, 82, 193)-Net over F₁₆ — Digital

Digital (35, 82, 193)-net over F₁₆, using

t-expansion [i] based on digital (33, 82, 193)-net over F₁₆, using
- net from sequence [i] based on digital (33, 192)-sequence over F₁₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 33 and N(F) ≥ 193, using
    - a curve from the manYPoints database [i]

(35, 82, 10926)-Net in Base 16 — Upper bound on s

There is no (35, 82, 10927)-net in base 16, because

1 times m-reduction [i] would yield (35, 81, 10927)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 34 243133 546617 134936 315825 351054 622434 146674 202164 899805 284363 384040 183133 162963 529633 637516 327216 > 16⁸¹ [i]