MinT - Best Known (49−27, 49, s)-Nets in Base 16

Best Known (49−27, 49, s)-Nets in Base 16

(49−27, 49, 103)-Net over F₁₆ — Constructive and digital

Digital (22, 49, 103)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (3, 16, 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, 33, 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]

(49−27, 49, 131)-Net over F₁₆ — Digital

Digital (22, 49, 131)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(16⁴⁹, 131, F₁₆, 2, 27) (dual of [(131, 2), 213, 28]-NRT-code), using
- constructionÂ X applied to AG(2;F,228P) âŠ‚ AG(2;F,232P) [i] based on
  1. linear OOA(16⁴⁶, 128, F₁₆, 2, 27) (dual of [(128, 2), 210, 28]-NRT-code), using algebraic-geometric NRT-code AG(2;F,228P) [i] based on function field F/F₁₆ with g(F) = 19 and N(F) ≥ 129, using
    - a curve from the manYPoints database [i]
  2. linear OOA(16⁴², 128, F₁₆, 2, 23) (dual of [(128, 2), 214, 24]-NRT-code), using algebraic-geometric NRT-code AG(2;F,232P) [i] based on function field F/F₁₆ with g(F) = 19 and N(F) ≥ 129 (see above)
  3. linear OOA(16³, 3, F₁₆, 2, 3) (dual of [(3, 2), 3, 4]-NRT-code), using
    - discarding factors / shortening the dual code based on linear OOA(16³, 16, F₁₆, 2, 3) (dual of [(16, 2), 29, 4]-NRT-code), using
      - Reedâ€“Solomon NRT-code RS(2;29,16) [i]

(49−27, 49, 150)-Net in Base 16 — Constructive

(22, 49, 150)-net in base 16, using

base change [i] based on digital (1, 28, 150)-net over F₁₂₈, using
- net from sequence [i] based on digital (1, 149)-sequence over F₁₂₈, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 1 and N(F) ≥ 150, using
    - a function field by SÃ©mirat [i]

(49−27, 49, 10544)-Net in Base 16 — Upper bound on s

There is no (22, 49, 10545)-net in base 16, because

1 times m-reduction [i] would yield (22, 48, 10545)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 6283 277606 214941 123124 172021 038492 931742 963786 970898 961776 > 16⁴⁸ [i]