MinT - Best Known (15, 15+25, s)-Nets in Base 256

Best Known (15, 15+25, s)-Nets in Base 256

(15, 15+25, 517)-Net over F₂₅₆ — Constructive and digital

Digital (15, 40, 517)-net over F₂₅₆, using

(u,Â u+v)-construction [i] based on
1. digital (1, 13, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]
2. digital (2, 27, 259)-net over F₂₅₆, using
  - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(15, 15+25, 610)-Net over F₂₅₆ — Digital

Digital (15, 40, 610)-net over F₂₅₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(256⁴⁰, 610, F₂₅₆, 4, 25) (dual of [(610, 4), 2400, 26]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(256¹³, 289, F₂₅₆, 4, 12) (dual of [(289, 4), 1143, 13]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(4;F,1143P) [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(256²⁷, 321, F₂₅₆, 4, 25) (dual of [(321, 4), 1257, 26]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(4;F,1258P) [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]

(15, 15+25, 1391872)-Net in Base 256 — Upper bound on s

There is no (15, 40, 1391873)-net in base 256, because

1 times m-reduction [i] would yield (15, 39, 1391873)-net in base 256, but
- the generalized Rao bound for nets shows that 256^mÂ â‰¥ 8343 750224 276948 417789 545100 323887 384265 676367 021650 944657 057818 726487 538286 046450 559747 438656 > 256³⁹ [i]