MinT - Best Known (43−26, 43, s)-Nets in Base 256

Best Known (43−26, 43, s)-Nets in Base 256

(43−26, 43, 518)-Net over F₂₅₆ — Constructive and digital

Digital (17, 43, 518)-net over F₂₅₆, using

(u,Â u+v)-construction [i] based on
1. digital (2, 15, 259)-net over F₂₅₆, using
  - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]
2. digital (2, 28, 259)-net over F₂₅₆, using
  - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆ (see above)

(43−26, 43, 642)-Net over F₂₅₆ — Digital

Digital (17, 43, 642)-net over F₂₅₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(256⁴³, 642, F₂₅₆, 2, 26) (dual of [(642, 2), 1241, 27]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(256¹⁵, 321, F₂₅₆, 2, 13) (dual of [(321, 2), 627, 14]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(2;F,628P) [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(256²⁸, 321, F₂₅₆, 2, 26) (dual of [(321, 2), 614, 27]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(2;F,615P) [i] based on function field F/F₂₅₆ with g(F) = 2 and N(F) ≥ 321 (see above)

(43−26, 43, 2053910)-Net in Base 256 — Upper bound on s

There is no (17, 43, 2053911)-net in base 256, because

the generalized Rao bound for nets shows that 256^mÂ â‰¥ 35 836016 932798 992869 423908 186922 614412 082465 846048 950616 810486 519332 078279 738126 514499 656449 835571 476916 > 256⁴³ [i]