MinT - Best Known (105−67, 105, s)-Nets in Base 16

Best Known (105−67, 105, s)-Nets in Base 16

(105−67, 105, 71)-Net over F₁₆ — Constructive and digital

Digital (38, 105, 71)-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 (3, 70, 38)-net over F₁₆, using
  - net from sequence [i] based on digital (3, 37)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 3 and N(F) ≥ 38, using
      - a function field by SÃ©mirat [i]

(105−67, 105, 120)-Net in Base 16 — Constructive

(38, 105, 120)-net in base 16, using

t-expansion [i] based on (37, 105, 120)-net in base 16, using
- 25 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]

(105−67, 105, 208)-Net over F₁₆ — Digital

Digital (38, 105, 208)-net over F₁₆, using

t-expansion [i] based on digital (37, 105, 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]

(105−67, 105, 5452)-Net in Base 16 — Upper bound on s

There is no (38, 105, 5453)-net in base 16, because

1 times m-reduction [i] would yield (38, 104, 5453)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 169438 349760 886008 229376 257572 727159 657874 914846 182680 380912 596933 695114 610606 801541 669639 672541 800409 028316 007596 462584 224536 > 16¹⁰⁴ [i]