MinT - Best Known (109−83, 109, s)-Nets in Base 16

Best Known (109−83, 109, s)-Nets in Base 16

(109−83, 109, 65)-Net over F₁₆ — Constructive and digital

Digital (26, 109, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 109, 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]

(109−83, 109, 66)-Net in Base 16 — Constructive

(26, 109, 66)-net in base 16, using

t-expansion [i] based on (25, 109, 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]

(109−83, 109, 150)-Net over F₁₆ — Digital

Digital (26, 109, 150)-net over F₁₆, using

net from sequence [i] based on digital (26, 149)-sequence over F₁₆, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 26 and N(F) ≥ 150, using
  - a curve from the manYPoints database [i]

(109−83, 109, 1575)-Net in Base 16 — Upper bound on s

There is no (26, 109, 1576)-net in base 16, because

1 times m-reduction [i] would yield (26, 108, 1576)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 11200 558422 405908 656167 490792 589062 145437 900611 807088 952598 573704 521340 875516 558950 544475 347051 631004 064497 169158 209915 670588 615616 > 16¹⁰⁸ [i]