MinT - Best Known (114−87, 114, s)-Nets in Base 16

Best Known (114−87, 114, s)-Nets in Base 16

(114−87, 114, 65)-Net over F₁₆ — Constructive and digital

Digital (27, 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]

(114−87, 114, 66)-Net in Base 16 — Constructive

(27, 114, 66)-net in base 16, using

t-expansion [i] based on (25, 114, 66)-net in base 16, using
- net from sequence [i] based on (25, 65)-sequence in base 16, using
  - base expansion [i] based on digital (50, 65)-sequence over F₄, using
    - t-expansion [i] based on digital (49, 65)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 49 and N(F) ≥ 66, using
        T₆ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(114−87, 114, 156)-Net over F₁₆ — Digital

Digital (27, 114, 156)-net over F₁₆, using

net from sequence [i] based on digital (27, 155)-sequence over F₁₆, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 27 and N(F) ≥ 156, using
  - a curve from the manYPoints database [i]

(114−87, 114, 1619)-Net in Base 16 — Upper bound on s

There is no (27, 114, 1620)-net in base 16, because

1 times m-reduction [i] would yield (27, 113, 1620)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 11805 081202 735589 682951 026500 472504 802823 114722 248848 475033 760901 118136 259362 835302 874863 243324 381797 416517 570843 764762 114386 300753 877276 > 16¹¹³ [i]