MinT - Best Known (106−69, 106, s)-Nets in Base 16

Best Known (106−69, 106, s)-Nets in Base 16

(106−69, 106, 65)-Net over F₁₆ — Constructive and digital

Digital (37, 106, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 106, 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]

(106−69, 106, 120)-Net in Base 16 — Constructive

(37, 106, 120)-net in base 16, using

24 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]

(106−69, 106, 208)-Net over F₁₆ — Digital

Digital (37, 106, 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]

(106−69, 106, 4701)-Net in Base 16 — Upper bound on s

There is no (37, 106, 4702)-net in base 16, because

1 times m-reduction [i] would yield (37, 105, 4702)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 2 712102 786043 278915 736367 621109 573064 389295 940464 265569 587217 670732 788299 838854 599043 099984 464272 966600 364695 495904 265848 823196 > 16¹⁰⁵ [i]