MinT - Best Known (22, 22+40, s)-Nets in Base 256

Best Known (22, 22+40, s)-Nets in Base 256

(22, 22+40, 516)-Net over F₂₅₆ — Constructive and digital

Digital (22, 62, 516)-net over F₂₅₆, using

(u,Â u+v)-construction [i] based on
1. digital (1, 21, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]
2. digital (1, 41, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆ (see above)

(22, 22+40, 578)-Net over F₂₅₆ — Digital

Digital (22, 62, 578)-net over F₂₅₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(256⁶², 578, F₂₅₆, 6, 40) (dual of [(578, 6), 3406, 41]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(256²¹, 289, F₂₅₆, 6, 20) (dual of [(289, 6), 1713, 21]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(6;F,1713P) [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(256⁴¹, 289, F₂₅₆, 6, 40) (dual of [(289, 6), 1693, 41]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(6;F,1693P) [i] based on function field F/F₂₅₆ with g(F) = 1 and N(F) ≥ 289 (see above)

(22, 22+40, 951273)-Net in Base 256 — Upper bound on s

There is no (22, 62, 951274)-net in base 256, because

the generalized Rao bound for nets shows that 256^mÂ â‰¥ 204587 449096 882460 297778 027646 137119 424661 908859 180655 513635 055734 162967 644987 789814 061131 488225 178954 215582 998876 950993 703022 698072 736134 942379 969276 > 256⁶² [i]