MinT - Best Known (49, 49+39, s)-Nets in Base 16

Best Known (49, 49+39, s)-Nets in Base 16

(49, 49+39, 524)-Net over F₁₆ — Constructive and digital

Digital (49, 88, 524)-net over F₁₆, using

trace code for nets [i] based on digital (5, 44, 262)-net over F₂₅₆, using
- net from sequence [i] based on digital (5, 261)-sequence over F₂₅₆, using
  - Niederreiter sequence [i]

(49, 49+39, 646)-Net over F₁₆ — Digital

Digital (49, 88, 646)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(16⁸⁸, 646, F₁₆, 2, 39) (dual of [(646, 2), 1204, 40]-NRT-code), using
- trace code [i] based on linear OOA(256⁴⁴, 323, F₂₅₆, 2, 39) (dual of [(323, 2), 602, 40]-NRT-code), using
  - constructionÂ X applied to AG(2;F,600P) âŠ‚ AG(2;F,604P) [i] based on
    1. 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, using
      - sharp bound on the number of rational points on curves with genus 2 [i]
    2. linear OOA(256³⁷, 320, F₂₅₆, 2, 35) (dual of [(320, 2), 603, 36]-NRT-code), using algebraic-geometric NRT-code AG(2;F,604P) [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]

(49, 49+39, 172464)-Net in Base 16 — Upper bound on s

There is no (49, 88, 172465)-net in base 16, because

1 times m-reduction [i] would yield (49, 87, 172465)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 573 388488 904646 250554 707326 283893 604818 256838 618537 660384 986641 158100 704879 944083 334676 202675 815237 015776 > 16⁸⁷ [i]