MinT - Best Known (108−69, 108, s)-Nets in Base 16

Best Known (108−69, 108, s)-Nets in Base 16

(108−69, 108, 71)-Net over F₁₆ — Constructive and digital

Digital (39, 108, 71)-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 (3, 72, 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]

(108−69, 108, 120)-Net in Base 16 — Constructive

(39, 108, 120)-net in base 16, using

t-expansion [i] based on (37, 108, 120)-net in base 16, using
- 22 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]

(108−69, 108, 208)-Net over F₁₆ — Digital

Digital (39, 108, 208)-net over F₁₆, using

t-expansion [i] based on digital (37, 108, 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]

(108−69, 108, 5537)-Net in Base 16 — Upper bound on s

There is no (39, 108, 5538)-net in base 16, because

1 times m-reduction [i] would yield (39, 107, 5538)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 693 185740 362832 771072 259745 176388 071231 045536 762089 377199 219912 859711 798231 210205 283421 461004 846159 977829 703150 586185 833861 005856 > 16¹⁰⁷ [i]