MinT - Best Known (13, 13+21, s)-Nets in Base 256

Best Known (13, 13+21, s)-Nets in Base 256

(13, 13+21, 517)-Net over F₂₅₆ — Constructive and digital

Digital (13, 34, 517)-net over F₂₅₆, using

(u,Â u+v)-construction [i] based on
1. digital (1, 11, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]
2. digital (2, 23, 259)-net over F₂₅₆, using
  - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(13, 13+21, 610)-Net over F₂₅₆ — Digital

Digital (13, 34, 610)-net over F₂₅₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(256³⁴, 610, F₂₅₆, 3, 21) (dual of [(610, 3), 1796, 22]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(256¹¹, 289, F₂₅₆, 3, 10) (dual of [(289, 3), 856, 11]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(3;F,856P) [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(256²³, 321, F₂₅₆, 3, 21) (dual of [(321, 3), 940, 22]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(3;F,941P) [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]

(13, 13+21, 1572630)-Net in Base 256 — Upper bound on s

There is no (13, 34, 1572631)-net in base 256, because

1 times m-reduction [i] would yield (13, 33, 1572631)-net in base 256, but
- the generalized Rao bound for nets shows that 256^mÂ â‰¥ 29 642827 718251 391103 120908 805386 271608 370739 562298 615793 581272 500770 540654 203926 > 256³³ [i]