MinT - Best Known (127−97, 127, s)-Nets in Base 16

Best Known (127−97, 127, s)-Nets in Base 16

(127−97, 127, 65)-Net over F₁₆ — Constructive and digital

Digital (30, 127, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 127, 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]

(127−97, 127, 66)-Net in Base 16 — Constructive

(30, 127, 66)-net in base 16, using

t-expansion [i] based on (25, 127, 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]

(127−97, 127, 162)-Net over F₁₆ — Digital

Digital (30, 127, 162)-net over F₁₆, using

net from sequence [i] based on digital (30, 161)-sequence over F₁₆, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 30 and N(F) ≥ 162, using
  - a curve from the manYPoints database [i]

(127−97, 127, 1782)-Net in Base 16 — Upper bound on s

There is no (30, 127, 1783)-net in base 16, because

1 times m-reduction [i] would yield (30, 126, 1783)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 52 780217 404826 405615 404019 988011 503700 484897 137399 344695 608059 738750 573329 371919 323010 744970 055479 874648 980475 609726 318493 617194 678639 736137 422293 751811 > 16¹²⁶ [i]