MinT - Best Known (36, 36+65, s)-Nets in Base 16

Best Known (36, 36+65, s)-Nets in Base 16

(36, 36+65, 66)-Net over F₁₆ — Constructive and digital

Digital (36, 101, 66)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (2, 34, 33)-net over F₁₆, using
  - net from sequence [i] based on digital (2, 32)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 2 and N(F) ≥ 33, using
      - a function field by SÃ©mirat [i]
2. digital (2, 67, 33)-net over F₁₆, using
  - net from sequence [i] based on digital (2, 32)-sequence over F₁₆ (see above)

(36, 36+65, 120)-Net in Base 16 — Constructive

(36, 101, 120)-net in base 16, using

24 times m-reduction [i] based on (36, 125, 120)-net in base 16, using
- base change [i] based on digital (11, 100, 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]

(36, 36+65, 193)-Net over F₁₆ — Digital

Digital (36, 101, 193)-net over F₁₆, using

t-expansion [i] based on digital (33, 101, 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]

(36, 36+65, 4921)-Net in Base 16 — Upper bound on s

There is no (36, 101, 4922)-net in base 16, because

1 times m-reduction [i] would yield (36, 100, 4922)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 2 587825 726313 092986 839772 890744 423496 509152 381547 014385 403669 415130 832082 918795 997874 401067 303208 134053 370825 524385 038611 > 16¹⁰⁰ [i]