MinT - Best Known (112−85, 112, s)-Nets in Base 16

Best Known (112−85, 112, s)-Nets in Base 16

(112−85, 112, 65)-Net over F₁₆ — Constructive and digital

Digital (27, 112, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 112, 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]

(112−85, 112, 66)-Net in Base 16 — Constructive

(27, 112, 66)-net in base 16, using

t-expansion [i] based on (25, 112, 66)-net in base 16, using
- net from sequence [i] based on (25, 65)-sequence in base 16, using
  - base expansion [i] based on digital (50, 65)-sequence over F₄, using
    - t-expansion [i] based on digital (49, 65)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 49 and N(F) ≥ 66, using
        T₆ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(112−85, 112, 156)-Net over F₁₆ — Digital

Digital (27, 112, 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]

(112−85, 112, 1651)-Net in Base 16 — Upper bound on s

There is no (27, 112, 1652)-net in base 16, because

1 times m-reduction [i] would yield (27, 111, 1652)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 45 621708 310323 618830 068617 300202 353046 884715 418039 578830 268531 205313 552258 864130 712180 140269 199320 154314 978729 046358 261541 469238 308186 > 16¹¹¹ [i]