MinT - Best Known (18, 18+29, s)-Nets in Base 256

Best Known (18, 18+29, s)-Nets in Base 256

(18, 18+29, 518)-Net over F₂₅₆ — Constructive and digital

Digital (18, 47, 518)-net over F₂₅₆, using

(u,Â u+v)-construction [i] based on
1. digital (2, 16, 259)-net over F₂₅₆, using
  - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]
2. digital (2, 31, 259)-net over F₂₅₆, using
  - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆ (see above)

(18, 18+29, 642)-Net over F₂₅₆ — Digital

Digital (18, 47, 642)-net over F₂₅₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(256⁴⁷, 642, F₂₅₆, 3, 29) (dual of [(642, 3), 1879, 30]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(256¹⁶, 321, F₂₅₆, 3, 14) (dual of [(321, 3), 947, 15]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(3;F,948P) [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(256³¹, 321, F₂₅₆, 3, 29) (dual of [(321, 3), 932, 30]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(3;F,933P) [i] based on function field F/F₂₅₆ with g(F) = 2 and N(F) ≥ 321 (see above)

(18, 18+29, 1939565)-Net in Base 256 — Upper bound on s

There is no (18, 47, 1939566)-net in base 256, because

1 times m-reduction [i] would yield (18, 46, 1939566)-net in base 256, but
- the generalized Rao bound for nets shows that 256^mÂ â‰¥ 601 227167 364998 986233 785249 601119 465837 246608 219997 967903 922017 588660 780505 628417 834001 076434 357712 357524 314896 > 256⁴⁶ [i]