MinT - Best Known (69, 69+61, s)-Nets in Base 16

Best Known (69, 69+61, s)-Nets in Base 16

(69, 69+61, 522)-Net over F₁₆ — Constructive and digital

Digital (69, 130, 522)-net over F₁₆, using

trace code for nets [i] based on digital (4, 65, 261)-net over F₂₅₆, using
- net from sequence [i] based on digital (4, 260)-sequence over F₂₅₆, using
  - Niederreiter sequence [i]

(69, 69+61, 644)-Net over F₁₆ — Digital

Digital (69, 130, 644)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(16¹³⁰, 644, F₁₆, 2, 61) (dual of [(644, 2), 1158, 62]-NRT-code), using
- 2 times NRT-code embedding in larger space [i] based on linear OOA(16¹²⁶, 642, F₁₆, 2, 61) (dual of [(642, 2), 1158, 62]-NRT-code), using
  - extracting embedded OOA [i] based on digital (65, 126, 642)-net over F₁₆, using
    - trace code for nets [i] based on digital (2, 63, 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]

(69, 69+61, 120879)-Net in Base 16 — Upper bound on s

There is no (69, 130, 120880)-net in base 16, because

1 times m-reduction [i] would yield (69, 129, 120880)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 214530 713572 960870 282183 395010 487873 018650 471499 371059 747653 336811 583927 465492 269554 046641 831924 352493 503800 218238 731222 022747 759152 775869 100553 035905 039126 > 16¹²⁹ [i]