MinT - Best Known (98−44, 98, s)-Nets in Base 16

Best Known (98−44, 98, s)-Nets in Base 16

(98−44, 98, 524)-Net over F₁₆ — Constructive and digital

Digital (54, 98, 524)-net over F₁₆, using

trace code for nets [i] based on digital (5, 49, 262)-net over F₂₅₆, using
- net from sequence [i] based on digital (5, 261)-sequence over F₂₅₆, using
  - Niederreiter sequence [i]

(98−44, 98, 646)-Net over F₁₆ — Digital

Digital (54, 98, 646)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(16⁹⁸, 646, F₁₆, 2, 44) (dual of [(646, 2), 1194, 45]-NRT-code), using
- trace code [i] based on linear OOA(256⁴⁹, 323, F₂₅₆, 2, 44) (dual of [(323, 2), 597, 45]-NRT-code), using
  - constructionÂ X applied to AG(2;F,595P) âŠ‚ AG(2;F,599P) [i] based on
    1. linear OOA(256⁴⁶, 320, F₂₅₆, 2, 44) (dual of [(320, 2), 594, 45]-NRT-code), using algebraic-geometric NRT-code AG(2;F,595P) [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(256⁴², 320, F₂₅₆, 2, 40) (dual of [(320, 2), 598, 41]-NRT-code), using algebraic-geometric NRT-code AG(2;F,599P) [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]

(98−44, 98, 139486)-Net in Base 16 — Upper bound on s

There is no (54, 98, 139487)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 10087 949095 248640 486315 670540 548163 109984 878607 802240 085349 245503 167105 935309 738779 064801 977440 628369 352446 426898 399761 > 16⁹⁸ [i]