MinT - Best Known (14, 40, s)-Nets in Base 256

Best Known (14, 40, s)-Nets in Base 256

(14, 40, 515)-Net over F₂₅₆ — Constructive and digital

Digital (14, 40, 515)-net over F₂₅₆, using

(u,Â u+v)-construction [i] based on
1. digital (0, 13, 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]
2. digital (1, 27, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(14, 40, 546)-Net over F₂₅₆ — Digital

Digital (14, 40, 546)-net over F₂₅₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(256⁴⁰, 546, F₂₅₆, 6, 26) (dual of [(546, 6), 3236, 27]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(256¹³, 257, F₂₅₆, 6, 13) (dual of [(257, 6), 1529, 14]-NRT-code), using
    - extended Reedâ€“Solomon NRT-code RS_e(6;1529,256) [i]
  2. linear OOA(256²⁷, 289, F₂₅₆, 6, 26) (dual of [(289, 6), 1707, 27]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(6;F,1707P) [i] based on function field F/F₂₅₆ with g(F) = 1 and N(F) ≥ 289, using
      - sharp bound on the number of rational points on curves with genus 1 [i]

(14, 40, 571255)-Net in Base 256 — Upper bound on s

There is no (14, 40, 571256)-net in base 256, because

the generalized Rao bound for nets shows that 256^mÂ â‰¥ 2 136003 210856 489707 356165 538888 845761 593456 814525 538048 659236 844116 398048 765555 081859 245080 548466 > 256⁴⁰ [i]