MinT - Best Known (51, 51+41, s)-Nets in Base 16

Best Known (51, 51+41, s)-Nets in Base 16

(51, 51+41, 524)-Net over F₁₆ — Constructive and digital

Digital (51, 92, 524)-net over F₁₆, using

trace code for nets [i] based on digital (5, 46, 262)-net over F₂₅₆, using
- net from sequence [i] based on digital (5, 261)-sequence over F₂₅₆, using
  - Niederreiter sequence [i]

(51, 51+41, 646)-Net over F₁₆ — Digital

Digital (51, 92, 646)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(16⁹², 646, F₁₆, 2, 41) (dual of [(646, 2), 1200, 42]-NRT-code), using
- trace code [i] based on linear OOA(256⁴⁶, 323, F₂₅₆, 2, 41) (dual of [(323, 2), 600, 42]-NRT-code), using
  - constructionÂ X applied to AG(2;F,598P) âŠ‚ AG(2;F,602P) [i] based on
    1. linear OOA(256⁴³, 320, F₂₅₆, 2, 41) (dual of [(320, 2), 597, 42]-NRT-code), using algebraic-geometric NRT-code AG(2;F,598P) [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(256³⁹, 320, F₂₅₆, 2, 37) (dual of [(320, 2), 601, 38]-NRT-code), using algebraic-geometric NRT-code AG(2;F,602P) [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]

(51, 51+41, 166698)-Net in Base 16 — Upper bound on s

There is no (51, 92, 166699)-net in base 16, because

1 times m-reduction [i] would yield (51, 91, 166699)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 37 577241 965676 340288 977457 084180 460698 688627 590469 551689 910635 745593 166528 764847 913958 536302 623141 860910 594076 > 16⁹¹ [i]