MinT - Best Known (23−14, 23, s)-Nets in Base 128

Best Known (23−14, 23, s)-Nets in Base 128

(23−14, 23, 300)-Net over F₁₂₈ — Constructive and digital

Digital (9, 23, 300)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. digital (1, 8, 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, 15, 150)-net over F₁₂₈, using
  - net from sequence [i] based on digital (1, 149)-sequence over F₁₂₈ (see above)

(23−14, 23, 301)-Net over F₁₂₈ — Digital

Digital (9, 23, 301)-net over F₁₂₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(128²³, 301, F₁₂₈, 2, 14) (dual of [(301, 2), 579, 15]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(128⁷, 129, F₁₂₈, 2, 7) (dual of [(129, 2), 251, 8]-NRT-code), using
    - extended Reedâ€“Solomon NRT-code RS_e(2;251,128) [i]
  2. linear OOA(128¹⁶, 172, F₁₂₈, 2, 14) (dual of [(172, 2), 328, 15]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(2;F,329P) [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]

(23−14, 23, 386)-Net in Base 128 — Constructive

(9, 23, 386)-net in base 128, using

(u,Â u+v)-construction [i] based on
1. digital (0, 7, 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, 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]

(23−14, 23, 223253)-Net in Base 128 — Upper bound on s

There is no (9, 23, 223254)-net in base 128, because

the generalized Rao bound for nets shows that 128^mÂ â‰¥ 2 923020 445821 915534 032741 645898 649211 568258 442696 > 128²³ [i]