MinT - Best Known (123−87, 123, s)-Nets in Base 16

Best Known (123−87, 123, s)-Nets in Base 16

(123−87, 123, 65)-Net over F₁₆ — Constructive and digital

Digital (36, 123, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 123, 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]

(123−87, 123, 120)-Net in Base 16 — Constructive

(36, 123, 120)-net in base 16, using

2 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]

(123−87, 123, 193)-Net over F₁₆ — Digital

Digital (36, 123, 193)-net over F₁₆, using

t-expansion [i] based on digital (33, 123, 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]

(123−87, 123, 2911)-Net in Base 16 — Upper bound on s

There is no (36, 123, 2912)-net in base 16, because

1 times m-reduction [i] would yield (36, 122, 2912)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 800 081710 882001 113547 474885 600433 739263 089579 499277 656352 571490 476588 006825 191730 280964 305575 429666 190779 360032 513284 942338 690363 023287 762198 266266 > 16¹²² [i]