MinT - Best Known (53−21, 53, s)-Nets in Base 16

Best Known (53−21, 53, s)-Nets in Base 16

Digital (32, 53, 538)-net over F₁₆, using

Digital (32, 53, 871)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16⁵³, 871, F₁₆, 21) (dual of [871, 818, 22]-code), using
- 222 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (3, 0, 0, 0, 1, 12 times 0, 1, 34 times 0, 1, 68 times 0, 1, 100 times 0) [i] based on linear OA(16⁴⁶, 642, F₁₆, 21) (dual of [642, 596, 22]-code), using
  - trace code [i] based on linear OA(256²³, 321, F₂₅₆, 21) (dual of [321, 298, 22]-code), using
    - extended algebraic-geometric code AG_e(F,299P) [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 (32, 53, 551195)-net in base 16, because

1 times m-reduction [i] would yield (32, 52, 551195)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 411 379797 203691 867523 954570 084792 097829 465221 385547 279847 661751 > 16⁵² [i]