MinT - Best Known (38, 38+69, s)-Nets in Base 16

Best Known (38, 38+69, s)-Nets in Base 16

(38, 38+69, 66)-Net over F₁₆ — Constructive and digital

Digital (38, 107, 66)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (2, 36, 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, 71, 33)-net over F₁₆, using
  - net from sequence [i] based on digital (2, 32)-sequence over F₁₆ (see above)

(38, 38+69, 120)-Net in Base 16 — Constructive

(38, 107, 120)-net in base 16, using

t-expansion [i] based on (37, 107, 120)-net in base 16, using
- 23 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]

(38, 38+69, 208)-Net over F₁₆ — Digital

Digital (38, 107, 208)-net over F₁₆, using

t-expansion [i] based on digital (37, 107, 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]

(38, 38+69, 5102)-Net in Base 16 — Upper bound on s

There is no (38, 107, 5103)-net in base 16, because

1 times m-reduction [i] would yield (38, 106, 5103)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 43 367216 640968 411092 711720 982285 783214 725537 234739 801792 750910 133621 506163 274923 035365 139814 068480 212854 948774 029299 845452 717381 > 16¹⁰⁶ [i]