MinT - Best Known (39, 39+82, s)-Nets in Base 16

Best Known (39, 39+82, s)-Nets in Base 16

(39, 39+82, 65)-Net over F₁₆ — Constructive and digital

Digital (39, 121, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 121, 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]

(39, 39+82, 120)-Net in Base 16 — Constructive

(39, 121, 120)-net in base 16, using

t-expansion [i] based on (37, 121, 120)-net in base 16, using
- 9 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+82, 208)-Net over F₁₆ — Digital

Digital (39, 121, 208)-net over F₁₆, using

t-expansion [i] based on digital (37, 121, 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+82, 3827)-Net in Base 16 — Upper bound on s

There is no (39, 121, 3828)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 50 401110 001187 152365 990276 948810 482204 292363 333018 016747 308674 365374 655788 249832 535543 612923 996055 798041 469026 991128 374997 453264 383300 260838 976846 > 16¹²¹ [i]