MinT - Best Known (42, 42+33, s)-Nets in Base 16

Best Known (42, 42+33, s)-Nets in Base 16

(42, 42+33, 522)-Net over F₁₆ — Constructive and digital

Digital (42, 75, 522)-net over F₁₆, using

1 times m-reduction [i] based on digital (42, 76, 522)-net over F₁₆, using
- trace code for nets [i] based on digital (4, 38, 261)-net over F₂₅₆, using
  - net from sequence [i] based on digital (4, 260)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(42, 42+33, 644)-Net over F₁₆ — Digital

Digital (42, 75, 644)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(16⁷⁵, 644, F₁₆, 2, 33) (dual of [(644, 2), 1213, 34]-NRT-code), using
- 16¹ times duplication [i] based on linear OOA(16⁷⁴, 644, F₁₆, 2, 33) (dual of [(644, 2), 1214, 34]-NRT-code), using
  - 2 times NRT-code embedding in larger space [i] based on linear OOA(16⁷⁰, 642, F₁₆, 2, 33) (dual of [(642, 2), 1214, 34]-NRT-code), using
    - extracting embedded OOA [i] based on digital (37, 70, 642)-net over F₁₆, using
      - trace code for nets [i] based on digital (2, 35, 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]

(42, 42+33, 168066)-Net in Base 16 — Upper bound on s

There is no (42, 75, 168067)-net in base 16, because

1 times m-reduction [i] would yield (42, 74, 168067)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 127325 060979 501746 136934 081601 173025 708968 465424 191419 532320 664370 394363 934645 904847 286456 > 16⁷⁴ [i]