MinT - Best Known (126−94, 126, s)-Nets in Base 16

Best Known (126−94, 126, s)-Nets in Base 16

(126−94, 126, 65)-Net over F₁₆ — Constructive and digital

Digital (32, 126, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 126, 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]

(126−94, 126, 76)-Net in Base 16 — Constructive

(32, 126, 76)-net in base 16, using

t-expansion [i] based on (31, 126, 76)-net in base 16, using
- 4 times m-reduction [i] based on (31, 130, 76)-net in base 16, using
  - base change [i] based on digital (5, 104, 76)-net over F₃₂, using
    - net from sequence [i] based on digital (5, 75)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 5 and N(F) ≥ 76, using
        a function field by SÃ©mirat [i]

(126−94, 126, 168)-Net over F₁₆ — Digital

Digital (32, 126, 168)-net over F₁₆, using

t-expansion [i] based on digital (31, 126, 168)-net over F₁₆, using
- net from sequence [i] based on digital (31, 167)-sequence over F₁₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 31 and N(F) ≥ 168, using
    - a curve from the manYPoints database [i]

(126−94, 126, 2044)-Net in Base 16 — Upper bound on s

There is no (32, 126, 2045)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 53 039537 031741 722089 914671 275088 387968 855518 334784 235674 706586 903166 234503 359161 720112 519507 543021 180018 775991 304611 086373 308845 650340 779762 809667 688976 > 16¹²⁶ [i]