MinT - Best Known (61−39, 61, s)-Nets in Base 256

Best Known (61−39, 61, s)-Nets in Base 256

(61−39, 61, 517)-Net over F₂₅₆ — Constructive and digital

Digital (22, 61, 517)-net over F₂₅₆, using

(u,Â u+v)-construction [i] based on
1. digital (1, 20, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]
2. digital (2, 41, 259)-net over F₂₅₆, using
  - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(61−39, 61, 610)-Net over F₂₅₆ — Digital

Digital (22, 61, 610)-net over F₂₅₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(256⁶¹, 610, F₂₅₆, 5, 39) (dual of [(610, 5), 2989, 40]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(256²⁰, 289, F₂₅₆, 5, 19) (dual of [(289, 5), 1425, 20]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(5;F,1425P) [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]
  2. linear OOA(256⁴¹, 321, F₂₅₆, 5, 39) (dual of [(321, 5), 1564, 40]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(5;F,1565P) [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]

(61−39, 61, 1252104)-Net in Base 256 — Upper bound on s

There is no (22, 61, 1252105)-net in base 256, because

1 times m-reduction [i] would yield (22, 60, 1252105)-net in base 256, but
- the generalized Rao bound for nets shows that 256^mÂ â‰¥ 3 121784 038166 339939 153950 461859 949001 451563 623301 943853 415346 263686 080184 151225 746623 792986 897447 551942 540657 478373 951727 963688 384043 937142 990976 > 256⁶⁰ [i]