MinT - Best Known (56−22, 56, s)-Nets in Base 16

Best Known (56−22, 56, s)-Nets in Base 16

Digital (34, 56, 538)-net over F₁₆, using

Digital (34, 56, 952)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16⁵⁶, 952, F₁₆, 22) (dual of [952, 896, 23]-code), using
- 302 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (3, 0, 0, 1, 11 times 0, 1, 27 times 0, 1, 56 times 0, 1, 88 times 0, 1, 112 times 0) [i] based on linear OA(16⁴⁸, 642, F₁₆, 22) (dual of [642, 594, 23]-code), using
  - trace code [i] based on linear OA(256²⁴, 321, F₂₅₆, 22) (dual of [321, 297, 23]-code), using
    - extended algebraic-geometric code AG_e(F,298P) [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 (34, 56, 441553)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 26 960093 786343 196959 748709 844912 886763 044973 685434 636514 372285 480496 > 16⁵⁶ [i]