MinT - Best Known (16, 16+25, s)-Nets in Base 256

Best Known (16, 16+25, s)-Nets in Base 256

(16, 16+25, 518)-Net over F₂₅₆ — Constructive and digital

Digital (16, 41, 518)-net over F₂₅₆, using

(u,Â u+v)-construction [i] based on
1. digital (2, 14, 259)-net over F₂₅₆, using
  - net from sequence [i] based on digital (2, 258)-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 (see above)

(16, 16+25, 642)-Net over F₂₅₆ — Digital

Digital (16, 41, 642)-net over F₂₅₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(256⁴¹, 642, F₂₅₆, 2, 25) (dual of [(642, 2), 1243, 26]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(256¹⁴, 321, F₂₅₆, 2, 12) (dual of [(321, 2), 628, 13]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(2;F,629P) [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(256²⁷, 321, F₂₅₆, 2, 25) (dual of [(321, 2), 615, 26]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(2;F,616P) [i] based on function field F/F₂₅₆ with g(F) = 2 and N(F) ≥ 321 (see above)

(16, 16+25, 2209462)-Net in Base 256 — Upper bound on s

There is no (16, 41, 2209463)-net in base 256, because

1 times m-reduction [i] would yield (16, 40, 2209463)-net in base 256, but
- the generalized Rao bound for nets shows that 256^mÂ â‰¥ 2 135989 352639 035084 184830 202982 130406 017619 346983 629959 094078 897123 814181 543876 487969 276485 136306 > 256⁴⁰ [i]