MinT - Best Known (32, 32+65, s)-Nets in Base 16

Best Known (32, 32+65, s)-Nets in Base 16

(32, 32+65, 65)-Net over F₁₆ — Constructive and digital

Digital (32, 97, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 97, 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]

(32, 32+65, 120)-Net in Base 16 — Constructive

(32, 97, 120)-net in base 16, using

8 times m-reduction [i] based on (32, 105, 120)-net in base 16, using
- base change [i] based on digital (11, 84, 120)-net over F₃₂, using
  - net from sequence [i] based on digital (11, 119)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 11 and N(F) ≥ 120, using
      - a function field by SÃ©mirat [i]

(32, 32+65, 168)-Net over F₁₆ — Digital

Digital (32, 97, 168)-net over F₁₆, using

t-expansion [i] based on digital (31, 97, 168)-net over F₁₆, using
- net from sequence [i] based on digital (31, 167)-sequence over F₁₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 31 and N(F) ≥ 168, using
    - a curve from the manYPoints database [i]

(32, 32+65, 3474)-Net in Base 16 — Upper bound on s

There is no (32, 97, 3475)-net in base 16, because

1 times m-reduction [i] would yield (32, 96, 3475)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 39 405401 079739 743875 117288 140405 143811 620170 432375 067378 389613 478215 666301 049678 518511 149887 149328 064644 494136 994876 > 16⁹⁶ [i]