MinT - Best Known (124−93, 124, s)-Nets in Base 16

Best Known (124−93, 124, s)-Nets in Base 16

(124−93, 124, 65)-Net over F₁₆ — Constructive and digital

Digital (31, 124, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 124, 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]

(124−93, 124, 76)-Net in Base 16 — Constructive

(31, 124, 76)-net in base 16, using

6 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]

(124−93, 124, 168)-Net over F₁₆ — Digital

Digital (31, 124, 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]

(124−93, 124, 1964)-Net in Base 16 — Upper bound on s

There is no (31, 124, 1965)-net in base 16, because

1 times m-reduction [i] would yield (31, 123, 1965)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 12934 370615 201606 868969 056277 805046 369558 907811 592839 415307 081937 412450 572544 404577 401214 426169 567249 835362 449570 260333 055703 253899 730317 701869 878976 > 16¹²³ [i]