MinT - Best Known (52, 52+42, s)-Nets in Base 16

Best Known (52, 52+42, s)-Nets in Base 16

(52, 52+42, 524)-Net over F₁₆ — Constructive and digital

Digital (52, 94, 524)-net over F₁₆, using

trace code for nets [i] based on digital (5, 47, 262)-net over F₂₅₆, using
- net from sequence [i] based on digital (5, 261)-sequence over F₂₅₆, using
  - Niederreiter sequence [i]

(52, 52+42, 646)-Net over F₁₆ — Digital

Digital (52, 94, 646)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(16⁹⁴, 646, F₁₆, 2, 42) (dual of [(646, 2), 1198, 43]-NRT-code), using
- trace code [i] based on linear OOA(256⁴⁷, 323, F₂₅₆, 2, 42) (dual of [(323, 2), 599, 43]-NRT-code), using
  - constructionÂ X applied to AG(2;F,597P) âŠ‚ AG(2;F,601P) [i] based on
    1. linear OOA(256⁴⁴, 320, F₂₅₆, 2, 42) (dual of [(320, 2), 596, 43]-NRT-code), using algebraic-geometric NRT-code AG(2;F,597P) [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(256⁴⁰, 320, F₂₅₆, 2, 38) (dual of [(320, 2), 600, 39]-NRT-code), using algebraic-geometric NRT-code AG(2;F,601P) [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]

(52, 52+42, 141983)-Net in Base 16 — Upper bound on s

There is no (52, 94, 141984)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 153933 200176 918692 500269 076125 075962 224943 719470 753359 733888 886575 150648 194830 341469 723221 736561 566741 423369 926211 > 16⁹⁴ [i]