MinT - Best Known (113−52, 113, s)-Nets in Base 16

Best Known (113−52, 113, s)-Nets in Base 16

(113−52, 113, 522)-Net over F₁₆ — Constructive and digital

Digital (61, 113, 522)-net over F₁₆, using

1 times m-reduction [i] based on digital (61, 114, 522)-net over F₁₆, using
- trace code for nets [i] based on digital (4, 57, 261)-net over F₂₅₆, using
  - net from sequence [i] based on digital (4, 260)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(113−52, 113, 644)-Net over F₁₆ — Digital

Digital (61, 113, 644)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(16¹¹³, 644, F₁₆, 2, 52) (dual of [(644, 2), 1175, 53]-NRT-code), using
- 16¹ times duplication [i] based on linear OOA(16¹¹², 644, F₁₆, 2, 52) (dual of [(644, 2), 1176, 53]-NRT-code), using
  - 2 times NRT-code embedding in larger space [i] based on linear OOA(16¹⁰⁸, 642, F₁₆, 2, 52) (dual of [(642, 2), 1176, 53]-NRT-code), using
    - extracting embedded OOA [i] based on digital (56, 108, 642)-net over F₁₆, using
      - trace code for nets [i] based on digital (2, 54, 321)-net over F₂₅₆, using
        net from sequence [i] based on digital (2, 320)-sequence over F₂₅₆, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [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]

(113−52, 113, 120348)-Net in Base 16 — Upper bound on s

There is no (61, 113, 120349)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 11630 373388 501736 295494 826847 541873 914417 686952 611933 619739 514004 851548 289853 886561 263265 333797 495666 601435 489263 285349 723812 563317 732986 > 16¹¹³ [i]