MinT - Best Known (116−91, 116, s)-Nets in Base 16

Best Known (116−91, 116, s)-Nets in Base 16

(116−91, 116, 65)-Net over F₁₆ — Constructive and digital

Digital (25, 116, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 116, 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]

(116−91, 116, 66)-Net in Base 16 — Constructive

(25, 116, 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]

(116−91, 116, 144)-Net over F₁₆ — Digital

Digital (25, 116, 144)-net over F₁₆, using

net from sequence [i] based on digital (25, 143)-sequence over F₁₆, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 25 and N(F) ≥ 144, using
  - a curve from the manYPoints database [i]

(116−91, 116, 1378)-Net in Base 16 — Upper bound on s

There is no (25, 116, 1379)-net in base 16, because

1 times m-reduction [i] would yield (25, 115, 1379)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 2 994251 974011 528518 708367 407281 237801 150293 052325 350192 855284 902834 492117 391223 987136 341963 279924 005905 310002 095430 459557 212112 088454 364576 > 16¹¹⁵ [i]