MinT - Best Known (97−59, 97, s)-Nets in Base 16

Best Known (97−59, 97, s)-Nets in Base 16

(97−59, 97, 103)-Net over F₁₆ — Constructive and digital

Digital (38, 97, 103)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (3, 32, 38)-net over F₁₆, using
  - net from sequence [i] based on digital (3, 37)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 3 and N(F) ≥ 38, using
      - a function field by SÃ©mirat [i]
2. digital (6, 65, 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]

(97−59, 97, 128)-Net in Base 16 — Constructive

(38, 97, 128)-net in base 16, using

2 times m-reduction [i] based on (38, 99, 128)-net in base 16, using
- base change [i] based on digital (5, 66, 128)-net over F₆₄, using
  - net from sequence [i] based on digital (5, 127)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 5 and N(F) ≥ 128, using
      - a function field by SÃ©mirat [i]

(97−59, 97, 208)-Net over F₁₆ — Digital

Digital (38, 97, 208)-net over F₁₆, using

t-expansion [i] based on digital (37, 97, 208)-net over F₁₆, using
- net from sequence [i] based on digital (37, 207)-sequence over F₁₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 37 and N(F) ≥ 208, using
    - Drinfeld modules of rank 1 [i]

(97−59, 97, 7519)-Net in Base 16 — Upper bound on s

There is no (38, 97, 7520)-net in base 16, because

1 times m-reduction [i] would yield (38, 96, 7520)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 39 542845 942906 846583 647569 458895 933199 602673 582987 568426 802428 360108 302240 290627 867396 339962 107179 843458 229637 263451 > 16⁹⁶ [i]