MinT - Best Known (53, 53+37, s)-Nets in Base 16

Best Known (53, 53+37, s)-Nets in Base 16

(53, 53+37, 530)-Net over F₁₆ — Constructive and digital

Digital (53, 90, 530)-net over F₁₆, using

trace code for nets [i] based on digital (8, 45, 265)-net over F₂₅₆, using
- net from sequence [i] based on digital (8, 264)-sequence over F₂₅₆, using
  - Niederreiter sequence [i]

(53, 53+37, 1026)-Net over F₁₆ — Digital

Digital (53, 90, 1026)-net over F₁₆, using

trace code for nets [i] based on digital (8, 45, 513)-net over F₂₅₆, using
- net from sequence [i] based on digital (8, 512)-sequence over F₂₅₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 8 and N(F) ≥ 513, using
    - K_1,1 from the tower of function fields by Niederreiter and Xing based on the tower by GarcÃa and Stichtenoth over F₂₅₆ [i]

(53, 53+37, 452620)-Net in Base 16 — Upper bound on s

There is no (53, 90, 452621)-net in base 16, because

1 times m-reduction [i] would yield (53, 89, 452621)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 146789 086210 651022 353403 594462 668495 094735 520389 871146 389776 849087 909652 869701 836347 022346 468894 402886 820096 > 16⁸⁹ [i]