MinT - Best Known (57−23, 57, s)-Nets in Base 16

Best Known (57−23, 57, s)-Nets in Base 16

Digital (34, 57, 531)-net over F₁₆, using

Digital (34, 57, 811)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16⁵⁷, 811, F₁₆, 23) (dual of [811, 754, 24]-code), using
- 162 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (3, 0, 0, 1, 8 times 0, 1, 23 times 0, 1, 47 times 0, 1, 77 times 0) [i] based on linear OA(16⁵⁰, 642, F₁₆, 23) (dual of [642, 592, 24]-code), using
  - trace code [i] based on linear OA(256²⁵, 321, F₂₅₆, 23) (dual of [321, 296, 24]-code), using
    - extended algebraic-geometric code AG_e(F,297P) [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, 57, 441553)-net in base 16, because

1 times m-reduction [i] would yield (34, 56, 441553)-net in base 16, but
- 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]