MinT - Best Known (98−61, 98, s)-Nets in Base 16

Best Known (98−61, 98, s)-Nets in Base 16

(98−61, 98, 89)-Net over F₁₆ — Constructive and digital

Digital (37, 98, 89)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (1, 31, 24)-net over F₁₆, using
  - net from sequence [i] based on digital (1, 23)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 1 and N(F) ≥ 24, using
      - a function field by SÃ©mirat [i]
2. digital (6, 67, 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]

(98−61, 98, 120)-Net in Base 16 — Constructive

(37, 98, 120)-net in base 16, using

32 times m-reduction [i] based on (37, 130, 120)-net in base 16, using
- base change [i] based on digital (11, 104, 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]

(98−61, 98, 208)-Net over F₁₆ — Digital

Digital (37, 98, 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]

(98−61, 98, 6264)-Net in Base 16 — Upper bound on s

There is no (37, 98, 6265)-net in base 16, because

1 times m-reduction [i] would yield (37, 97, 6265)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 632 347861 381171 398646 606589 877282 611476 249751 357642 618148 516789 518801 933436 544729 227937 878857 460857 676833 931686 667376 > 16⁹⁷ [i]