MinT - Best Known (51−33, 51, s)-Nets in Base 256

Best Known (51−33, 51, s)-Nets in Base 256

(51−33, 51, 516)-Net over F₂₅₆ — Constructive and digital

Digital (18, 51, 516)-net over F₂₅₆, using

(u,Â u+v)-construction [i] based on
1. digital (1, 17, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]
2. digital (1, 34, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆ (see above)

(51−33, 51, 578)-Net over F₂₅₆ — Digital

Digital (18, 51, 578)-net over F₂₅₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(256⁵¹, 578, F₂₅₆, 6, 33) (dual of [(578, 6), 3417, 34]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(256¹⁷, 289, F₂₅₆, 6, 16) (dual of [(289, 6), 1717, 17]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(6;F,1717P) [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(256³⁴, 289, F₂₅₆, 6, 33) (dual of [(289, 6), 1700, 34]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(6;F,1700P) [i] based on function field F/F₂₅₆ with g(F) = 1 and N(F) ≥ 289 (see above)

(51−33, 51, 894838)-Net in Base 256 — Upper bound on s

There is no (18, 51, 894839)-net in base 256, because

1 times m-reduction [i] would yield (18, 50, 894839)-net in base 256, but
- the generalized Rao bound for nets shows that 256^mÂ â‰¥ 2 582263 049744 587878 109833 001452 691332 151203 983878 926276 525709 631400 197887 994599 621938 925888 373010 829393 624132 834881 576871 > 256⁵⁰ [i]