MinT - Best Known (10, 10+18, s)-Nets in Base 256

Best Known (10, 10+18, s)-Nets in Base 256

(10, 10+18, 515)-Net over F₂₅₆ — Constructive and digital

Digital (10, 28, 515)-net over F₂₅₆, using

(u,Â u+v)-construction [i] based on
1. digital (0, 9, 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, 19, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(10, 10+18, 546)-Net over F₂₅₆ — Digital

Digital (10, 28, 546)-net over F₂₅₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(256²⁸, 546, F₂₅₆, 4, 18) (dual of [(546, 4), 2156, 19]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(256⁹, 257, F₂₅₆, 4, 9) (dual of [(257, 4), 1019, 10]-NRT-code), using
    - extended Reedâ€“Solomon NRT-code RS_e(4;1019,256) [i]
  2. linear OOA(256¹⁹, 289, F₂₅₆, 4, 18) (dual of [(289, 4), 1137, 19]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(4;F,1137P) [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]

(10, 10+18, 505254)-Net in Base 256 — Upper bound on s

There is no (10, 28, 505255)-net in base 256, because

the generalized Rao bound for nets shows that 256^mÂ â‰¥ 26 960341 420696 499919 310274 445174 000802 440001 217674 717819 227143 559976 > 256²⁸ [i]