MinT - Best Known (39, 39+70, s)-Nets in Base 16

Best Known (39, 39+70, s)-Nets in Base 16

(39, 39+70, 66)-Net over F₁₆ — Constructive and digital

Digital (39, 109, 66)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (2, 37, 33)-net over F₁₆, using
  - net from sequence [i] based on digital (2, 32)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 2 and N(F) ≥ 33, using
      - a function field by SÃ©mirat [i]
2. digital (2, 72, 33)-net over F₁₆, using
  - net from sequence [i] based on digital (2, 32)-sequence over F₁₆ (see above)

(39, 39+70, 120)-Net in Base 16 — Constructive

(39, 109, 120)-net in base 16, using

t-expansion [i] based on (37, 109, 120)-net in base 16, using
- 21 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]

(39, 39+70, 208)-Net over F₁₆ — Digital

Digital (39, 109, 208)-net over F₁₆, using

t-expansion [i] based on digital (37, 109, 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]

(39, 39+70, 5194)-Net in Base 16 — Upper bound on s

There is no (39, 109, 5195)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 178183 292263 098352 043015 999125 312336 032127 943180 329645 630010 066907 480671 895271 748688 356551 031045 722690 926188 821800 813920 556999 447376 > 16¹⁰⁹ [i]