MinT - Best Known (111−73, 111, s)-Nets in Base 16

Best Known (111−73, 111, s)-Nets in Base 16

(111−73, 111, 65)-Net over F₁₆ — Constructive and digital

Digital (38, 111, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 111, 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]

(111−73, 111, 120)-Net in Base 16 — Constructive

(38, 111, 120)-net in base 16, using

t-expansion [i] based on (37, 111, 120)-net in base 16, using
- 19 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]

(111−73, 111, 208)-Net over F₁₆ — Digital

Digital (38, 111, 208)-net over F₁₆, using

t-expansion [i] based on digital (37, 111, 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]

(111−73, 111, 4528)-Net in Base 16 — Upper bound on s

There is no (38, 111, 4529)-net in base 16, because

1 times m-reduction [i] would yield (38, 110, 4529)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 2 839917 224605 367623 777632 041339 011844 038241 970670 550664 520132 635451 769023 096923 116208 471305 346596 508655 825508 310382 797752 705125 810036 > 16¹¹⁰ [i]