MinT - Best Known (18, 49, s)-Nets in Base 256

Best Known (18, 49, s)-Nets in Base 256

(18, 49, 517)-Net over F₂₅₆ — Constructive and digital

Digital (18, 49, 517)-net over F₂₅₆, using

(u,Â u+v)-construction [i] based on
1. digital (1, 16, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]
2. digital (2, 33, 259)-net over F₂₅₆, using
  - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(18, 49, 610)-Net over F₂₅₆ — Digital

Digital (18, 49, 610)-net over F₂₅₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(256⁴⁹, 610, F₂₅₆, 5, 31) (dual of [(610, 5), 3001, 32]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(256¹⁶, 289, F₂₅₆, 5, 15) (dual of [(289, 5), 1429, 16]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(5;F,1429P) [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(256³³, 321, F₂₅₆, 5, 31) (dual of [(321, 5), 1572, 32]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(5;F,1573P) [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]

(18, 49, 1281126)-Net in Base 256 — Upper bound on s

There is no (18, 49, 1281127)-net in base 256, because

1 times m-reduction [i] would yield (18, 48, 1281127)-net in base 256, but
- the generalized Rao bound for nets shows that 256^mÂ â‰¥ 39 402316 091537 482833 249077 235807 759632 858201 315326 106071 848997 358615 893465 874596 876558 486234 961670 878516 028399 311776 > 256⁴⁸ [i]