MinT - Best Known (113−42, 113, s)-Nets in Base 16

Best Known (113−42, 113, s)-Nets in Base 16

(113−42, 113, 581)-Net over F₁₆ — Constructive and digital

Digital (71, 113, 581)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (6, 27, 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]
2. digital (44, 86, 516)-net over F₁₆, using
  - trace code for nets [i] based on digital (1, 43, 258)-net over F₂₅₆, using
    - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]

(113−42, 113, 2262)-Net over F₁₆ — Digital

Digital (71, 113, 2262)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(113−42, 113, 1744659)-Net in Base 16 — Upper bound on s

There is no (71, 113, 1744660)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 11629 445480 357004 037658 301371 321278 074660 720581 076964 856403 645117 358863 908682 318049 717820 372298 742558 815360 371536 504138 302206 258640 419776 > 16¹¹³ [i]