MinT - Best Known (37, 37+66, s)-Nets in Base 16

Best Known (37, 37+66, s)-Nets in Base 16

(37, 37+66, 66)-Net over F₁₆ — Constructive and digital

Digital (37, 103, 66)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (2, 35, 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, 68, 33)-net over F₁₆, using
  - net from sequence [i] based on digital (2, 32)-sequence over F₁₆ (see above)

(37, 37+66, 120)-Net in Base 16 — Constructive

(37, 103, 120)-net in base 16, using

27 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]

(37, 37+66, 208)-Net over F₁₆ — Digital

Digital (37, 103, 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]

(37, 37+66, 5011)-Net in Base 16 — Upper bound on s

There is no (37, 103, 5012)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 10582 598328 455971 804423 001532 835578 366461 969500 302557 811487 788049 743205 924265 334903 162946 581096 027219 486156 825725 288866 502716 > 16¹⁰³ [i]