MinT - Best Known (127−103, 127, s)-Nets in Base 16

Best Known (127−103, 127, s)-Nets in Base 16

(127−103, 127, 65)-Net over F₁₆ — Constructive and digital

Digital (24, 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]

(127−103, 127, 129)-Net over F₁₆ — Digital

Digital (24, 127, 129)-net over F₁₆, using

t-expansion [i] based on digital (19, 127, 129)-net over F₁₆, using
- net from sequence [i] based on digital (19, 128)-sequence over F₁₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 19 and N(F) ≥ 129, using
    - a curve from the manYPoints database [i]

(127−103, 127, 1221)-Net in Base 16 — Upper bound on s

There is no (24, 127, 1222)-net in base 16, because

1 times m-reduction [i] would yield (24, 126, 1222)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 54 538776 914024 482971 918345 251104 923147 893967 051730 409483 060369 753516 717600 396228 013388 559410 103282 255638 280848 235363 042709 809306 351665 404273 773187 370456 > 16¹²⁶ [i]