MinT - Best Known (122−87, 122, s)-Nets in Base 16

Best Known (122−87, 122, s)-Nets in Base 16

(122−87, 122, 65)-Net over F₁₆ — Constructive and digital

Digital (35, 122, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 122, 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]

(122−87, 122, 104)-Net in Base 16 — Constructive

(35, 122, 104)-net in base 16, using

8 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]

(122−87, 122, 193)-Net over F₁₆ — Digital

Digital (35, 122, 193)-net over F₁₆, using

t-expansion [i] based on digital (33, 122, 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]

(122−87, 122, 2728)-Net in Base 16 — Upper bound on s

There is no (35, 122, 2729)-net in base 16, because

1 times m-reduction [i] would yield (35, 121, 2729)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 50 256585 168611 216511 657230 241922 674366 167449 204291 045104 761305 771066 921160 030974 183334 690308 370226 044660 426965 311931 116161 079438 236560 348434 019456 > 16¹²¹ [i]