MinT - Best Known (22, 37, s)-Nets in Base 16

Best Known (22, 37, s)-Nets in Base 16

Digital (22, 37, 531)-net over F₁₆, using

Digital (22, 37, 685)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16³⁷, 685, F₁₆, 15) (dual of [685, 648, 16]-code), using
- 40 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 7 times 0, 1, 31 times 0) [i] based on linear OA(16³⁴, 642, F₁₆, 15) (dual of [642, 608, 16]-code), using
  - trace code [i] based on linear OA(256¹⁷, 321, F₂₅₆, 15) (dual of [321, 304, 16]-code), using
    - extended algebraic-geometric code AG_e(F,305P) [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 (22, 37, 351108)-net in base 16, because

1 times m-reduction [i] would yield (22, 36, 351108)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 22 300950 188183 736616 200122 547185 575697 830816 > 16³⁶ [i]