MinT - Best Known (77−34, 77, s)-Nets in Base 16

Best Known (77−34, 77, s)-Nets in Base 16

(77−34, 77, 522)-Net over F₁₆ — Constructive and digital

Digital (43, 77, 522)-net over F₁₆, using

1 times m-reduction [i] based on digital (43, 78, 522)-net over F₁₆, using
- trace code for nets [i] based on digital (4, 39, 261)-net over F₂₅₆, using
  - net from sequence [i] based on digital (4, 260)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(77−34, 77, 644)-Net over F₁₆ — Digital

Digital (43, 77, 644)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(16⁷⁷, 644, F₁₆, 2, 34) (dual of [(644, 2), 1211, 35]-NRT-code), using
- 16¹ times duplication [i] based on linear OOA(16⁷⁶, 644, F₁₆, 2, 34) (dual of [(644, 2), 1212, 35]-NRT-code), using
  - 2 times NRT-code embedding in larger space [i] based on linear OOA(16⁷², 642, F₁₆, 2, 34) (dual of [(642, 2), 1212, 35]-NRT-code), using
    - extracting embedded OOA [i] based on digital (38, 72, 642)-net over F₁₆, using
      - trace code for nets [i] based on digital (2, 36, 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]

(77−34, 77, 136075)-Net in Base 16 — Upper bound on s

There is no (43, 77, 136076)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 521 512225 318703 396793 864590 051248 479831 757366 198869 057796 770328 525217 159453 032289 920744 476856 > 16⁷⁷ [i]