MinT - Best Known (128−101, 128, s)-Nets in Base 16

Best Known (128−101, 128, s)-Nets in Base 16

(128−101, 128, 65)-Net over F₁₆ — Constructive and digital

Digital (27, 128, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 128, 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]

(128−101, 128, 66)-Net in Base 16 — Constructive

(27, 128, 66)-net in base 16, using

t-expansion [i] based on (25, 128, 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]

(128−101, 128, 156)-Net over F₁₆ — Digital

Digital (27, 128, 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]

(128−101, 128, 1458)-Net in Base 16 — Upper bound on s

There is no (27, 128, 1459)-net in base 16, because

1 times m-reduction [i] would yield (27, 127, 1459)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 856 513286 960071 722899 811221 137890 531549 177832 596999 902678 761154 962395 314315 043935 646438 999128 928170 267456 232880 204075 510376 626677 068505 155845 637617 825501 > 16¹²⁷ [i]