MinT - Best Known (118−83, 118, s)-Nets in Base 16

Best Known (118−83, 118, s)-Nets in Base 16

(118−83, 118, 65)-Net over F₁₆ — Constructive and digital

Digital (35, 118, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 118, 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]

(118−83, 118, 120)-Net in Base 16 — Constructive

(35, 118, 120)-net in base 16, using

2 times m-reduction [i] based on (35, 120, 120)-net in base 16, using
- base change [i] based on digital (11, 96, 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]

(118−83, 118, 193)-Net over F₁₆ — Digital

Digital (35, 118, 193)-net over F₁₆, using

t-expansion [i] based on digital (33, 118, 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]

(118−83, 118, 2914)-Net in Base 16 — Upper bound on s

There is no (35, 118, 2915)-net in base 16, because

1 times m-reduction [i] would yield (35, 117, 2915)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 764 813896 749259 508823 703491 944986 030309 018881 449655 263919 034218 754511 145952 181882 976109 762331 619464 808429 843843 764921 979229 789089 528460 788976 > 16¹¹⁷ [i]