MinT - Best Known (49−30, 49, s)-Nets in Base 256

Best Known (49−30, 49, s)-Nets in Base 256

(49−30, 49, 518)-Net over F₂₅₆ — Constructive and digital

Digital (19, 49, 518)-net over F₂₅₆, using

(u,Â u+v)-construction [i] based on
1. digital (2, 17, 259)-net over F₂₅₆, using
  - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]
2. digital (2, 32, 259)-net over F₂₅₆, using
  - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆ (see above)

(49−30, 49, 642)-Net over F₂₅₆ — Digital

Digital (19, 49, 642)-net over F₂₅₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(256⁴⁹, 642, F₂₅₆, 2, 30) (dual of [(642, 2), 1235, 31]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(256¹⁷, 321, F₂₅₆, 2, 15) (dual of [(321, 2), 625, 16]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(2;F,626P) [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]
  2. linear OOA(256³², 321, F₂₅₆, 2, 30) (dual of [(321, 2), 610, 31]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(2;F,611P) [i] based on function field F/F₂₅₆ with g(F) = 2 and N(F) ≥ 321 (see above)

(49−30, 49, 1854137)-Net in Base 256 — Upper bound on s

There is no (19, 49, 1854138)-net in base 256, because

the generalized Rao bound for nets shows that 256^mÂ â‰¥ 10086 923267 505174 628944 451875 003938 376256 454444 922793 246899 790001 445512 099881 585232 674470 510715 433017 887767 978260 797976 > 256⁴⁹ [i]