MinT - Best Known (34, 34+78, s)-Nets in Base 16

Best Known (34, 34+78, s)-Nets in Base 16

(34, 34+78, 65)-Net over F₁₆ — Constructive and digital

Digital (34, 112, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 112, 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]

(34, 34+78, 120)-Net in Base 16 — Constructive

(34, 112, 120)-net in base 16, using

3 times m-reduction [i] based on (34, 115, 120)-net in base 16, using
- base change [i] based on digital (11, 92, 120)-net over F₃₂, using
  - net from sequence [i] based on digital (11, 119)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 11 and N(F) ≥ 120, using
      - a function field by SÃ©mirat [i]

(34, 34+78, 193)-Net over F₁₆ — Digital

Digital (34, 112, 193)-net over F₁₆, using

t-expansion [i] based on digital (33, 112, 193)-net over F₁₆, using
- net from sequence [i] based on digital (33, 192)-sequence over F₁₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 33 and N(F) ≥ 193, using
    - a curve from the manYPoints database [i]

(34, 34+78, 2925)-Net in Base 16 — Upper bound on s

There is no (34, 112, 2926)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 736 243856 636340 071593 673643 890587 927626 993951 995716 127050 721470 277958 873621 254545 439750 667078 238610 432742 932859 228086 136085 047527 574436 > 16¹¹² [i]