MinT - Best Known (52−32, 52, s)-Nets in Base 256

Best Known (52−32, 52, s)-Nets in Base 256

(52−32, 52, 518)-Net over F₂₅₆ — Constructive and digital

Digital (20, 52, 518)-net over F₂₅₆, using

(u,Â u+v)-construction [i] based on
1. digital (2, 18, 259)-net over F₂₅₆, using
  - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]
2. digital (2, 34, 259)-net over F₂₅₆, using
  - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆ (see above)

(52−32, 52, 642)-Net over F₂₅₆ — Digital

Digital (20, 52, 642)-net over F₂₅₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(256⁵², 642, F₂₅₆, 3, 32) (dual of [(642, 3), 1874, 33]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(256¹⁸, 321, F₂₅₆, 3, 16) (dual of [(321, 3), 945, 17]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(3;F,946P) [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(256³⁴, 321, F₂₅₆, 3, 32) (dual of [(321, 3), 929, 33]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(3;F,930P) [i] based on function field F/F₂₅₆ with g(F) = 2 and N(F) ≥ 321 (see above)

(52−32, 52, 1789684)-Net in Base 256 — Upper bound on s

There is no (20, 52, 1789685)-net in base 256, because

the generalized Rao bound for nets shows that 256^mÂ â‰¥ 169230 339858 411013 076197 091431 432615 647403 446899 123103 908522 528330 402016 862181 963919 554465 112863 934661 802163 947692 142915 198551 > 256⁵² [i]