MinT - Best Known (21−13, 21, s)-Nets in Base 128

Best Known (21−13, 21, s)-Nets in Base 128

(21−13, 21, 300)-Net over F₁₂₈ — Constructive and digital

Digital (8, 21, 300)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. digital (1, 7, 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]
2. digital (1, 14, 150)-net over F₁₂₈, using
  - net from sequence [i] based on digital (1, 149)-sequence over F₁₂₈ (see above)

(21−13, 21, 301)-Net over F₁₂₈ — Digital

Digital (8, 21, 301)-net over F₁₂₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(128²¹, 301, F₁₂₈, 2, 13) (dual of [(301, 2), 581, 14]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(128⁶, 129, F₁₂₈, 2, 6) (dual of [(129, 2), 252, 7]-NRT-code), using
    - extended Reedâ€“Solomon NRT-code RS_e(2;252,128) [i]
  2. linear OOA(128¹⁵, 172, F₁₂₈, 2, 13) (dual of [(172, 2), 329, 14]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(2;F,330P) [i] based on function field F/F₁₂₈ with g(F) = 2 and N(F) ≥ 172, using
      - sharp bound on the number of rational points on curves with genus 2 [i]

(21−13, 21, 386)-Net in Base 128 — Constructive

(8, 21, 386)-net in base 128, using

(u,Â u+v)-construction [i] based on
1. digital (0, 6, 129)-net over F₁₂₈, using
  - net from sequence [i] based on digital (0, 128)-sequence over F₁₂₈, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 0 and N(F) ≥ 129, using
      - the rational function field F₁₂₈(x) [i]
    - Niederreiter sequence [i]
2. (2, 15, 257)-net in base 128, using
  - 1 times m-reduction [i] based on (2, 16, 257)-net in base 128, using
    - base change [i] based on digital (0, 14, 257)-net over F₂₅₆, using
      - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
        Niederreiter sequence [i]

(21−13, 21, 249142)-Net in Base 128 — Upper bound on s

There is no (8, 21, 249143)-net in base 128, because

1 times m-reduction [i] would yield (8, 20, 249143)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 1 393826 149870 220825 278848 254473 525907 270813 > 128²⁰ [i]