MinT - Best Known (41, 41+32, s)-Nets in Base 16

Best Known (41, 41+32, s)-Nets in Base 16

(41, 41+32, 522)-Net over F₁₆ — Constructive and digital

Digital (41, 73, 522)-net over F₁₆, using

1 times m-reduction [i] based on digital (41, 74, 522)-net over F₁₆, using
- trace code for nets [i] based on digital (4, 37, 261)-net over F₂₅₆, using
  - net from sequence [i] based on digital (4, 260)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(41, 41+32, 644)-Net over F₁₆ — Digital

Digital (41, 73, 644)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(16⁷³, 644, F₁₆, 2, 32) (dual of [(644, 2), 1215, 33]-NRT-code), using
- 16¹ times duplication [i] based on linear OOA(16⁷², 644, F₁₆, 2, 32) (dual of [(644, 2), 1216, 33]-NRT-code), using
  - 2 times NRT-code embedding in larger space [i] based on linear OOA(16⁶⁸, 642, F₁₆, 2, 32) (dual of [(642, 2), 1216, 33]-NRT-code), using
    - extracting embedded OOA [i] based on digital (36, 68, 642)-net over F₁₆, using
      - trace code for nets [i] based on digital (2, 34, 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]

(41, 41+32, 141324)-Net in Base 16 — Upper bound on s

There is no (41, 73, 141325)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 7957 298433 996586 495037 624434 983049 194518 018118 303684 500464 338310 075741 347091 261119 477376 > 16⁷³ [i]