MinT - Best Known (30−19, 30, s)-Nets in Base 128

Best Known (30−19, 30, s)-Nets in Base 128

(30−19, 30, 300)-Net over F₁₂₈ — Constructive and digital

Digital (11, 30, 300)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. digital (1, 10, 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, 20, 150)-net over F₁₂₈, using
  - net from sequence [i] based on digital (1, 149)-sequence over F₁₂₈ (see above)

(30−19, 30, 301)-Net over F₁₂₈ — Digital

Digital (11, 30, 301)-net over F₁₂₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(128³⁰, 301, F₁₂₈, 3, 19) (dual of [(301, 3), 873, 20]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(128⁹, 129, F₁₂₈, 3, 9) (dual of [(129, 3), 378, 10]-NRT-code), using
    - extended Reedâ€“Solomon NRT-code RS_e(3;378,128) [i]
  2. linear OOA(128²¹, 172, F₁₂₈, 3, 19) (dual of [(172, 3), 495, 20]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(3;F,496P) [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]

(30−19, 30, 321)-Net in Base 128

(11, 30, 321)-net in base 128, using

42 times m-reduction [i] based on (11, 72, 321)-net in base 128, using
- base change [i] based on digital (2, 63, 321)-net over F₂₅₆, using
  - net from sequence [i] based on digital (2, 320)-sequence over F₂₅₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 2 and N(F) ≥ 321, using
      - sharp bound on the number of rational points on curves with genus 2 [i]

(30−19, 30, 201297)-Net in Base 128 — Upper bound on s

There is no (11, 30, 201298)-net in base 128, because

1 times m-reduction [i] would yield (11, 29, 201298)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 12 855963 752541 907833 563644 203345 919080 887742 046550 072339 455314 > 128²⁹ [i]