MinT - Best Known (60−25, 60, s)-Nets in Base 16

Best Known (60−25, 60, s)-Nets in Base 16

Digital (35, 60, 524)-net over F₁₆, using

trace code for nets [i] based on digital (5, 30, 262)-net over F₂₅₆, using
- net from sequence [i] based on digital (5, 261)-sequence over F₂₅₆, using
  - Niederreiter sequence [i]

Digital (35, 60, 708)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16⁶⁰, 708, F₁₆, 25) (dual of [708, 648, 26]-code), using
- 60 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (3, 0, 1, 6 times 0, 1, 15 times 0, 1, 34 times 0) [i] based on linear OA(16⁵⁴, 642, F₁₆, 25) (dual of [642, 588, 26]-code), using
  - trace code [i] based on linear OA(256²⁷, 321, F₂₅₆, 25) (dual of [321, 294, 26]-code), using
    - extended algebraic-geometric code AG_e(F,295P) [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]

There is no (35, 60, 293439)-net in base 16, because

1 times m-reduction [i] would yield (35, 59, 293439)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 110429 228919 155924 129253 954249 586239 680074 666693 389217 335001 600423 582021 > 16⁵⁹ [i]