MinT - Best Known (28−17, 28, s)-Nets in Base 256

Best Known (28−17, 28, s)-Nets in Base 256

(28−17, 28, 517)-Net over F₂₅₆ — Constructive and digital

Digital (11, 28, 517)-net over F₂₅₆, using

(u,Â u+v)-construction [i] based on
1. digital (1, 9, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]
2. digital (2, 19, 259)-net over F₂₅₆, using
  - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(28−17, 28, 610)-Net over F₂₅₆ — Digital

Digital (11, 28, 610)-net over F₂₅₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(256²⁸, 610, F₂₅₆, 2, 17) (dual of [(610, 2), 1192, 18]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(256⁹, 289, F₂₅₆, 2, 8) (dual of [(289, 2), 569, 9]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(2;F,569P) [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(256¹⁹, 321, F₂₅₆, 2, 17) (dual of [(321, 2), 623, 18]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(2;F,624P) [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]

(28−17, 28, 1981339)-Net in Base 256 — Upper bound on s

There is no (11, 28, 1981340)-net in base 256, because

1 times m-reduction [i] would yield (11, 27, 1981340)-net in base 256, but
- the generalized Rao bound for nets shows that 256^mÂ â‰¥ 105312 317746 621689 346729 356441 709233 563114 739906 500599 439690 753851 > 256²⁷ [i]