MinT - Best Known (34, 34+38, s)-Nets in Base 16

Best Known (34, 34+38, s)-Nets in Base 16

(34, 34+38, 130)-Net over F₁₆ — Constructive and digital

Digital (34, 72, 130)-net over F₁₆, using

6 times m-reduction [i] based on digital (34, 78, 130)-net over F₁₆, using
- (u,Â u+v)-construction [i] based on
  1. digital (6, 28, 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 (6, 50, 65)-net over F₁₆, using
    - net from sequence [i] based on digital (6, 64)-sequence over F₁₆ (see above)

(34, 34+38, 192)-Net in Base 16 — Constructive

(34, 72, 192)-net in base 16, using

base change [i] based on (10, 48, 192)-net in base 64, using
- 1 times m-reduction [i] based on (10, 49, 192)-net in base 64, using
  - base change [i] based on digital (3, 42, 192)-net over F₁₂₈, using
    - net from sequence [i] based on digital (3, 191)-sequence over F₁₂₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 3 and N(F) ≥ 192, using
        a function field by SÃ©mirat [i]

(34, 34+38, 211)-Net over F₁₆ — Digital

Digital (34, 72, 211)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(34, 34+38, 225)-Net in Base 16

(34, 72, 225)-net in base 16, using

base change [i] based on digital (10, 48, 225)-net over F₆₄, using
- net from sequence [i] based on digital (10, 224)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 10 and N(F) ≥ 225, using
    - a curve from the manYPoints database [i]

(34, 34+38, 19314)-Net in Base 16 — Upper bound on s

There is no (34, 72, 19315)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 497 763575 603284 395179 003029 253348 002664 276381 479677 582771 160275 817340 222395 168050 041776 > 16⁷² [i]