MinT - Best Known (42−22, 42, s)-Nets in Base 16

Best Known (42−22, 42, s)-Nets in Base 16

(42−22, 42, 103)-Net over F₁₆ — Constructive and digital

Digital (20, 42, 103)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (3, 14, 38)-net over F₁₆, using
  - net from sequence [i] based on digital (3, 37)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 3 and N(F) ≥ 38, using
      - a function field by SÃ©mirat [i]
2. digital (6, 28, 65)-net over F₁₆, using
  - net from sequence [i] based on digital (6, 64)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 6 and N(F) ≥ 65, using
      - the Hermitian function field over F₁₆ [i]

(42−22, 42, 150)-Net in Base 16 — Constructive

(20, 42, 150)-net in base 16, using

t-expansion [i] based on (19, 42, 150)-net in base 16, using
- base change [i] based on digital (1, 24, 150)-net over F₁₂₈, using
  - net from sequence [i] based on digital (1, 149)-sequence over F₁₂₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 1 and N(F) ≥ 150, using
      - a function field by SÃ©mirat [i]

(42−22, 42, 155)-Net over F₁₆ — Digital

Digital (20, 42, 155)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(42−22, 42, 172)-Net in Base 16

(20, 42, 172)-net in base 16, using

base change [i] based on digital (2, 24, 172)-net over F₁₂₈, using
- net from sequence [i] based on digital (2, 171)-sequence over F₁₂₈, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 2 and N(F) ≥ 172, using
    - sharp bound on the number of rational points on curves with genus 2 [i]

(42−22, 42, 12950)-Net in Base 16 — Upper bound on s

There is no (20, 42, 12951)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 374 367900 260336 598278 772036 571299 875081 341563 737416 > 16⁴² [i]