MinT - Best Known (106−67, 106, s)-Nets in Base 16

Best Known (106−67, 106, s)-Nets in Base 16

(106−67, 106, 82)-Net over F₁₆ — Constructive and digital

Digital (39, 106, 82)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (0, 33, 17)-net over F₁₆, using
  - net from sequence [i] based on digital (0, 16)-sequence over F₁₆, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 0 and N(F) ≥ 17, using
      - the rational function field F₁₆(x) [i]
    - Niederreiter sequence [i]
2. digital (6, 73, 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]

(106−67, 106, 120)-Net in Base 16 — Constructive

(39, 106, 120)-net in base 16, using

t-expansion [i] based on (37, 106, 120)-net in base 16, using
- 24 times m-reduction [i] based on (37, 130, 120)-net in base 16, using
  - base change [i] based on digital (11, 104, 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]

(106−67, 106, 208)-Net over F₁₆ — Digital

Digital (39, 106, 208)-net over F₁₆, using

t-expansion [i] based on digital (37, 106, 208)-net over F₁₆, using
- net from sequence [i] based on digital (37, 207)-sequence over F₁₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 37 and N(F) ≥ 208, using
    - Drinfeld modules of rank 1 [i]

(106−67, 106, 5932)-Net in Base 16 — Upper bound on s

There is no (39, 106, 5933)-net in base 16, because

1 times m-reduction [i] would yield (39, 105, 5933)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 2 717955 622867 969270 579371 183017 916428 736592 290200 970020 624626 052213 166630 998988 955227 853756 903160 466579 005630 419579 861845 283636 > 16¹⁰⁵ [i]