MinT - Best Known (61, 61+45, s)-Nets in Base 16

Best Known (61, 61+45, s)-Nets in Base 16

(61, 61+45, 530)-Net over F₁₆ — Constructive and digital

Digital (61, 106, 530)-net over F₁₆, using

trace code for nets [i] based on digital (8, 53, 265)-net over F₂₅₆, using
- net from sequence [i] based on digital (8, 264)-sequence over F₂₅₆, using
  - Niederreiter sequence [i]

(61, 61+45, 1026)-Net over F₁₆ — Digital

Digital (61, 106, 1026)-net over F₁₆, using

trace code for nets [i] based on digital (8, 53, 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]

(61, 61+45, 337040)-Net in Base 16 — Upper bound on s

There is no (61, 106, 337041)-net in base 16, because

1 times m-reduction [i] would yield (61, 105, 337041)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 2 707699 675845 986840 745363 852467 696880 404517 912451 413676 650605 018120 140648 364432 399360 096233 684854 934351 649740 710543 831941 640256 > 16¹⁰⁵ [i]