MinT - Best Known (20, 56, s)-Nets in Base 256

Best Known (20, 56, s)-Nets in Base 256

(20, 56, 516)-Net over F₂₅₆ — Constructive and digital

Digital (20, 56, 516)-net over F₂₅₆, using

(u,Â u+v)-construction [i] based on
1. digital (1, 19, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]
2. digital (1, 37, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆ (see above)

(20, 56, 578)-Net over F₂₅₆ — Digital

Digital (20, 56, 578)-net over F₂₅₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(256⁵⁶, 578, F₂₅₆, 6, 36) (dual of [(578, 6), 3412, 37]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(256¹⁹, 289, F₂₅₆, 6, 18) (dual of [(289, 6), 1715, 19]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(6;F,1715P) [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(256³⁷, 289, F₂₅₆, 6, 36) (dual of [(289, 6), 1697, 37]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(6;F,1697P) [i] based on function field F/F₂₅₆ with g(F) = 1 and N(F) ≥ 289 (see above)

(20, 56, 920212)-Net in Base 256 — Upper bound on s

There is no (20, 56, 920213)-net in base 256, because

the generalized Rao bound for nets shows that 256^mÂ â‰¥ 726 851335 713705 030310 337904 544332 801337 616812 746014 897709 017382 419601 874687 088374 827730 026935 120369 874825 153021 872453 706291 387323 983246 > 256⁵⁶ [i]