MinT - Best Known (36, 129, s)-Nets in Base 16

Best Known (36, 129, s)-Nets in Base 16

(36, 129, 65)-Net over F₁₆ — Constructive and digital

Digital (36, 129, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 129, 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]

(36, 129, 104)-Net in Base 16 — Constructive

(36, 129, 104)-net in base 16, using

t-expansion [i] based on (35, 129, 104)-net in base 16, using
- 1 times m-reduction [i] based on (35, 130, 104)-net in base 16, using
  - base change [i] based on digital (9, 104, 104)-net over F₃₂, using
    - net from sequence [i] based on digital (9, 103)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 9 and N(F) ≥ 104, using
        a function field by SÃ©mirat [i]

(36, 129, 193)-Net over F₁₆ — Digital

Digital (36, 129, 193)-net over F₁₆, using

t-expansion [i] based on digital (33, 129, 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, 129, 2664)-Net in Base 16 — Upper bound on s

There is no (36, 129, 2665)-net in base 16, because

1 times m-reduction [i] would yield (36, 128, 2665)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 13551 768999 251228 011879 953845 909561 448729 397828 911385 250851 127362 982478 135328 158438 996265 662627 350785 187881 121578 911541 739740 283616 017280 250715 755433 586976 > 16¹²⁸ [i]