MinT - Best Known (111−82, 111, s)-Nets in Base 16

Best Known (111−82, 111, s)-Nets in Base 16

(111−82, 111, 65)-Net over F₁₆ — Constructive and digital

Digital (29, 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]

(111−82, 111, 76)-Net in Base 16 — Constructive

(29, 111, 76)-net in base 16, using

9 times m-reduction [i] based on (29, 120, 76)-net in base 16, using
- base change [i] based on digital (5, 96, 76)-net over F₃₂, using
  - net from sequence [i] based on digital (5, 75)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 5 and N(F) ≥ 76, using
      - a function field by SÃ©mirat [i]

(111−82, 111, 161)-Net over F₁₆ — Digital

Digital (29, 111, 161)-net over F₁₆, using

net from sequence [i] based on digital (29, 160)-sequence over F₁₆, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 29 and N(F) ≥ 161, using
  - a curve from the manYPoints database [i]

(111−82, 111, 1935)-Net in Base 16 — Upper bound on s

There is no (29, 111, 1936)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 46 313442 209851 862032 906992 734156 351828 932810 214161 290534 376445 401241 563552 846603 670111 870751 731194 973325 683841 560550 760056 354464 179516 > 16¹¹¹ [i]