MinT - Best Known (25−16, 25, s)-Nets in Base 256

Best Known (25−16, 25, s)-Nets in Base 256

(25−16, 25, 515)-Net over F₂₅₆ — Constructive and digital

Digital (9, 25, 515)-net over F₂₅₆, using

(u,Â u+v)-construction [i] based on
1. digital (0, 8, 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, 17, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(25−16, 25, 546)-Net over F₂₅₆ — Digital

Digital (9, 25, 546)-net over F₂₅₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(256²⁵, 546, F₂₅₆, 4, 16) (dual of [(546, 4), 2159, 17]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(256⁸, 257, F₂₅₆, 4, 8) (dual of [(257, 4), 1020, 9]-NRT-code), using
    - extended Reedâ€“Solomon NRT-code RS_e(4;1020,256) [i]
  2. linear OOA(256¹⁷, 289, F₂₅₆, 4, 16) (dual of [(289, 4), 1139, 17]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(4;F,1139P) [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]

(25−16, 25, 495332)-Net in Base 256 — Upper bound on s

There is no (9, 25, 495333)-net in base 256, because

the generalized Rao bound for nets shows that 256^mÂ â‰¥ 1 606955 273477 766165 923682 390028 740963 373001 974938 347338 276396 > 256²⁵ [i]