MinT - Best Known (57, 57+48, s)-Nets in Base 16

Best Known (57, 57+48, s)-Nets in Base 16

(57, 57+48, 522)-Net over F₁₆ — Constructive and digital

Digital (57, 105, 522)-net over F₁₆, using

1 times m-reduction [i] based on digital (57, 106, 522)-net over F₁₆, using
- trace code for nets [i] based on digital (4, 53, 261)-net over F₂₅₆, using
  - net from sequence [i] based on digital (4, 260)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(57, 57+48, 644)-Net over F₁₆ — Digital

Digital (57, 105, 644)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(16¹⁰⁵, 644, F₁₆, 2, 48) (dual of [(644, 2), 1183, 49]-NRT-code), using
- 16¹ times duplication [i] based on linear OOA(16¹⁰⁴, 644, F₁₆, 2, 48) (dual of [(644, 2), 1184, 49]-NRT-code), using
  - 2 times NRT-code embedding in larger space [i] based on linear OOA(16¹⁰⁰, 642, F₁₆, 2, 48) (dual of [(642, 2), 1184, 49]-NRT-code), using
    - extracting embedded OOA [i] based on digital (52, 100, 642)-net over F₁₆, using
      - trace code for nets [i] based on digital (2, 50, 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]

(57, 57+48, 121129)-Net in Base 16 — Upper bound on s

There is no (57, 105, 121130)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 2 708026 471302 345953 789737 450141 025627 368965 280075 292058 238046 689525 755045 006210 053056 079772 397327 448587 790312 692709 921903 401426 > 16¹⁰⁵ [i]