MinT - Best Known (28, 28+79, s)-Nets in Base 16

Best Known (28, 28+79, s)-Nets in Base 16

(28, 28+79, 65)-Net over F₁₆ — Constructive and digital

Digital (28, 107, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 107, 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]

(28, 28+79, 76)-Net in Base 16 — Constructive

(28, 107, 76)-net in base 16, using

8 times m-reduction [i] based on (28, 115, 76)-net in base 16, using
- base change [i] based on digital (5, 92, 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]

(28, 28+79, 156)-Net over F₁₆ — Digital

Digital (28, 107, 156)-net over F₁₆, using

t-expansion [i] based on digital (27, 107, 156)-net over F₁₆, using
- net from sequence [i] based on digital (27, 155)-sequence over F₁₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 27 and N(F) ≥ 156, using
    - a curve from the manYPoints database [i]

(28, 28+79, 1901)-Net in Base 16 — Upper bound on s

There is no (28, 107, 1902)-net in base 16, because

1 times m-reduction [i] would yield (28, 106, 1902)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 43 443079 561927 162214 947134 967186 184358 304345 718477 000660 145776 034606 452980 977698 519598 108740 985065 585019 465486 845185 271902 370596 > 16¹⁰⁶ [i]