MinT - Best Known (9, 9+14, s)-Nets in Base 256

Best Known (9, 9+14, s)-Nets in Base 256

(9, 9+14, 516)-Net over F₂₅₆ — Constructive and digital

Digital (9, 23, 516)-net over F₂₅₆, using

(u,Â u+v)-construction [i] based on
1. digital (1, 8, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]
2. digital (1, 15, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆ (see above)

(9, 9+14, 578)-Net over F₂₅₆ — Digital

Digital (9, 23, 578)-net over F₂₅₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(256²³, 578, F₂₅₆, 2, 14) (dual of [(578, 2), 1133, 15]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(256⁸, 289, F₂₅₆, 2, 7) (dual of [(289, 2), 570, 8]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(2;F,570P) [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(256¹⁵, 289, F₂₅₆, 2, 14) (dual of [(289, 2), 563, 15]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(2;F,563P) [i] based on function field F/F₂₅₆ with g(F) = 1 and N(F) ≥ 289 (see above)

(9, 9+14, 1084340)-Net in Base 256 — Upper bound on s

There is no (9, 23, 1084341)-net in base 256, because

the generalized Rao bound for nets shows that 256^mÂ â‰¥ 24 519994 511987 971655 339616 835805 892784 167087 201410 056336 > 256²³ [i]