MinT - Best Known (109−72, 109, s)-Nets in Base 16

Best Known (109−72, 109, s)-Nets in Base 16

(109−72, 109, 65)-Net over F₁₆ — Constructive and digital

Digital (37, 109, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 109, 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]

(109−72, 109, 120)-Net in Base 16 — Constructive

(37, 109, 120)-net in base 16, using

21 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]

(109−72, 109, 208)-Net over F₁₆ — Digital

Digital (37, 109, 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]

(109−72, 109, 4191)-Net in Base 16 — Upper bound on s

There is no (37, 109, 4192)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 177775 993009 555270 867481 457667 442873 524368 979521 574499 641595 428130 247310 022053 228600 868798 320312 188039 716563 457010 528010 525227 613306 > 16¹⁰⁹ [i]