MinT - Best Known (27−11, 27, s)-Nets in Base 16

Best Known (27−11, 27, s)-Nets in Base 16

Digital (16, 27, 531)-net over F₁₆, using

Digital (16, 27, 653)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16²⁷, 653, F₁₆, 11) (dual of [653, 626, 12]-code), using
- 10 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (1, 9 times 0) [i] based on linear OA(16²⁶, 642, F₁₆, 11) (dual of [642, 616, 12]-code), using
  - trace code [i] based on linear OA(256¹³, 321, F₂₅₆, 11) (dual of [321, 308, 12]-code), using
    - extended algebraic-geometric code AG_e(F,309P) [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 (16, 27, 317078)-net in base 16, because

1 times m-reduction [i] would yield (16, 26, 317078)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 20 282504 433454 735554 825872 606851 > 16²⁶ [i]