MinT - Best Known (101−46, 101, s)-Nets in Base 16

Best Known (101−46, 101, s)-Nets in Base 16

(101−46, 101, 522)-Net over F₁₆ — Constructive and digital

Digital (55, 101, 522)-net over F₁₆, using

1 times m-reduction [i] based on digital (55, 102, 522)-net over F₁₆, using
- trace code for nets [i] based on digital (4, 51, 261)-net over F₂₅₆, using
  - net from sequence [i] based on digital (4, 260)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(101−46, 101, 643)-Net over F₁₆ — Digital

Digital (55, 101, 643)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(16¹⁰¹, 643, F₁₆, 2, 46) (dual of [(643, 2), 1185, 47]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(16¹⁰¹, 644, F₁₆, 2, 46) (dual of [(644, 2), 1187, 47]-NRT-code), using
  - 16¹ times duplication [i] based on linear OOA(16¹⁰⁰, 644, F₁₆, 2, 46) (dual of [(644, 2), 1188, 47]-NRT-code), using
    - 2 times NRT-code embedding in larger space [i] based on linear OOA(16⁹⁶, 642, F₁₆, 2, 46) (dual of [(642, 2), 1188, 47]-NRT-code), using
      - extracting embedded OOA [i] based on digital (50, 96, 642)-net over F₁₆, using
        trace code for nets [i] based on digital (2, 48, 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]

(101−46, 101, 121892)-Net in Base 16 — Upper bound on s

There is no (55, 101, 121893)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 41 318365 802178 835847 002564 098971 411723 601518 708464 694056 437210 165601 039358 912575 187626 917529 459158 010687 517584 342960 719136 > 16¹⁰¹ [i]