MinT - Best Known (96−43, 96, s)-Nets in Base 16

Best Known (96−43, 96, s)-Nets in Base 16

(96−43, 96, 524)-Net over F₁₆ — Constructive and digital

Digital (53, 96, 524)-net over F₁₆, using

trace code for nets [i] based on digital (5, 48, 262)-net over F₂₅₆, using
- net from sequence [i] based on digital (5, 261)-sequence over F₂₅₆, using
  - Niederreiter sequence [i]

(96−43, 96, 646)-Net over F₁₆ — Digital

Digital (53, 96, 646)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(16⁹⁶, 646, F₁₆, 2, 43) (dual of [(646, 2), 1196, 44]-NRT-code), using
- trace code [i] based on linear OOA(256⁴⁸, 323, F₂₅₆, 2, 43) (dual of [(323, 2), 598, 44]-NRT-code), using
  - constructionÂ X applied to AG(2;F,596P) âŠ‚ AG(2;F,600P) [i] based on
    1. linear OOA(256⁴⁵, 320, F₂₅₆, 2, 43) (dual of [(320, 2), 595, 44]-NRT-code), using algebraic-geometric NRT-code AG(2;F,596P) [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(256⁴¹, 320, F₂₅₆, 2, 39) (dual of [(320, 2), 599, 40]-NRT-code), using algebraic-geometric NRT-code AG(2;F,600P) [i] based on function field F/F₂₅₆ with g(F) = 2 and N(F) ≥ 321 (see above)
    3. linear OOA(256³, 3, F₂₅₆, 2, 3) (dual of [(3, 2), 3, 4]-NRT-code), using
      - discarding factors / shortening the dual code based on linear OOA(256³, 256, F₂₅₆, 2, 3) (dual of [(256, 2), 509, 4]-NRT-code), using
        Reedâ€“Solomon NRT-code RS(2;509,256) [i]

(96−43, 96, 162024)-Net in Base 16 — Upper bound on s

There is no (53, 96, 162025)-net in base 16, because

1 times m-reduction [i] would yield (53, 95, 162025)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 2 462843 517282 556071 090086 175103 062057 183050 323673 635145 602092 862730 428859 510909 860865 145865 504755 609349 176152 175376 > 16⁹⁵ [i]