MinT - Best Known (19, 19+31, s)-Nets in Base 256

Best Known (19, 19+31, s)-Nets in Base 256

(19, 19+31, 518)-Net over F₂₅₆ — Constructive and digital

Digital (19, 50, 518)-net over F₂₅₆, using

(u,Â u+v)-construction [i] based on
1. digital (2, 17, 259)-net over F₂₅₆, using
  - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]
2. digital (2, 33, 259)-net over F₂₅₆, using
  - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆ (see above)

(19, 19+31, 642)-Net over F₂₅₆ — Digital

Digital (19, 50, 642)-net over F₂₅₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(256⁵⁰, 642, F₂₅₆, 3, 31) (dual of [(642, 3), 1876, 32]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(256¹⁷, 321, F₂₅₆, 3, 15) (dual of [(321, 3), 946, 16]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(3;F,947P) [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]
  2. linear OOA(256³³, 321, F₂₅₆, 3, 31) (dual of [(321, 3), 930, 32]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(3;F,931P) [i] based on function field F/F₂₅₆ with g(F) = 2 and N(F) ≥ 321 (see above)

(19, 19+31, 1854137)-Net in Base 256 — Upper bound on s

There is no (19, 50, 1854138)-net in base 256, because

1 times m-reduction [i] would yield (19, 49, 1854138)-net in base 256, but
- the generalized Rao bound for nets shows that 256^mÂ â‰¥ 10086 923267 505174 628944 451875 003938 376256 454444 922793 246899 790001 445512 099881 585232 674470 510715 433017 887767 978260 797976 > 256⁴⁹ [i]