MinT - Best Known (114−77, 114, s)-Nets in Base 16

Best Known (114−77, 114, s)-Nets in Base 16

(114−77, 114, 65)-Net over F₁₆ — Constructive and digital

Digital (37, 114, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 114, 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]

(114−77, 114, 120)-Net in Base 16 — Constructive

(37, 114, 120)-net in base 16, using

16 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]

(114−77, 114, 208)-Net over F₁₆ — Digital

Digital (37, 114, 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]

(114−77, 114, 3793)-Net in Base 16 — Upper bound on s

There is no (37, 114, 3794)-net in base 16, because

1 times m-reduction [i] would yield (37, 113, 3794)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 11742 471543 915547 388362 559775 871228 679195 116978 995304 849486 003807 988796 736913 099870 962484 294051 824985 046081 455551 124429 131403 675325 841456 > 16¹¹³ [i]