MinT - Best Known (20−13, 20, s)-Nets in Base 256

Best Known (20−13, 20, s)-Nets in Base 256

(20−13, 20, 515)-Net over F₂₅₆ — Constructive and digital

Digital (7, 20, 515)-net over F₂₅₆, using

(u,Â u+v)-construction [i] based on
1. digital (0, 6, 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, 14, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(20−13, 20, 546)-Net over F₂₅₆ — Digital

Digital (7, 20, 546)-net over F₂₅₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(256²⁰, 546, F₂₅₆, 3, 13) (dual of [(546, 3), 1618, 14]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(256⁶, 257, F₂₅₆, 3, 6) (dual of [(257, 3), 765, 7]-NRT-code), using
    - extended Reedâ€“Solomon NRT-code RS_e(3;765,256) [i]
  2. linear OOA(256¹⁴, 289, F₂₅₆, 3, 13) (dual of [(289, 3), 853, 14]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(3;F,853P) [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]

(20−13, 20, 496332)-Net in Base 256 — Upper bound on s

There is no (7, 20, 496333)-net in base 256, because

1 times m-reduction [i] would yield (7, 19, 496333)-net in base 256, but
- the generalized Rao bound for nets shows that 256^mÂ â‰¥ 5709 009200 821160 061230 269911 768002 868460 181116 > 256¹⁹ [i]