MinT - Best Known (31, 122, s)-Nets in Base 16

Best Known (31, 122, s)-Nets in Base 16

(31, 122, 65)-Net over F₁₆ — Constructive and digital

Digital (31, 122, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 122, 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]

(31, 122, 76)-Net in Base 16 — Constructive

(31, 122, 76)-net in base 16, using

8 times m-reduction [i] based on (31, 130, 76)-net in base 16, using
- base change [i] based on digital (5, 104, 76)-net over F₃₂, using
  - net from sequence [i] based on digital (5, 75)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 5 and N(F) ≥ 76, using
      - a function field by SÃ©mirat [i]

(31, 122, 168)-Net over F₁₆ — Digital

Digital (31, 122, 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]

(31, 122, 2006)-Net in Base 16 — Upper bound on s

There is no (31, 122, 2007)-net in base 16, because

1 times m-reduction [i] would yield (31, 121, 2007)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 50 365329 394610 330269 250240 163400 207662 672017 791869 550948 160347 751241 430890 071170 900354 201771 661342 269434 583551 574366 794015 889187 536022 987298 687976 > 16¹²¹ [i]