MinT - Best Known (81−36, 81, s)-Nets in Base 16

Best Known (81−36, 81, s)-Nets in Base 16

(81−36, 81, 522)-Net over F₁₆ — Constructive and digital

Digital (45, 81, 522)-net over F₁₆, using

1 times m-reduction [i] based on digital (45, 82, 522)-net over F₁₆, using
- trace code for nets [i] based on digital (4, 41, 261)-net over F₂₅₆, using
  - net from sequence [i] based on digital (4, 260)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(81−36, 81, 644)-Net over F₁₆ — Digital

Digital (45, 81, 644)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(16⁸¹, 644, F₁₆, 2, 36) (dual of [(644, 2), 1207, 37]-NRT-code), using
- 16¹ times duplication [i] based on linear OOA(16⁸⁰, 644, F₁₆, 2, 36) (dual of [(644, 2), 1208, 37]-NRT-code), using
  - 2 times NRT-code embedding in larger space [i] based on linear OOA(16⁷⁶, 642, F₁₆, 2, 36) (dual of [(642, 2), 1208, 37]-NRT-code), using
    - extracting embedded OOA [i] based on digital (40, 76, 642)-net over F₁₆, using
      - trace code for nets [i] based on digital (2, 38, 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]

(81−36, 81, 131991)-Net in Base 16 — Upper bound on s

There is no (45, 81, 131992)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 34 176931 678520 089263 041515 040056 743412 968143 191514 910110 764107 951462 211836 963748 321068 308402 865241 > 16⁸¹ [i]