MinT - Best Known (51−31, 51, s)-Nets in Base 256

Best Known (51−31, 51, s)-Nets in Base 256

(51−31, 51, 519)-Net over F₂₅₆ — Constructive and digital

Digital (20, 51, 519)-net over F₂₅₆, using

(u,Â u+v)-construction [i] based on
1. digital (2, 17, 259)-net over F₂₅₆, using
  - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]
2. digital (3, 34, 260)-net over F₂₅₆, using
  - net from sequence [i] based on digital (3, 259)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(51−31, 51, 642)-Net over F₂₅₆ — Digital

Digital (20, 51, 642)-net over F₂₅₆, using

1 times m-reduction [i] based on 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)

(51−31, 51, 2683439)-Net in Base 256 — Upper bound on s

There is no (20, 51, 2683440)-net in base 256, because

1 times m-reduction [i] would yield (20, 50, 2683440)-net in base 256, but
- the generalized Rao bound for nets shows that 256^mÂ â‰¥ 2 582251 684420 467797 701952 110983 396710 372286 313697 969916 464568 327559 820994 078546 752664 165748 544825 111125 698514 357970 804251 > 256⁵⁰ [i]