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

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

Digital (27, 43, 552)-net over F₁₆, using

Digital (27, 43, 1220)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16⁴³, 1220, F₁₆, 16) (dual of [1220, 1177, 17]-code), using
- 633 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (3, 0, 0, 0, 1, 13 times 0, 1, 37 times 0, 1, 80 times 0, 1, 126 times 0, 1, 165 times 0, 1, 202 times 0) [i] based on linear OA(16³⁴, 578, F₁₆, 16) (dual of [578, 544, 17]-code), using
  - trace code [i] based on linear OA(256¹⁷, 289, F₂₅₆, 16) (dual of [289, 272, 17]-code), using
    - extended algebraic-geometric code AG_e(F,272P) [i] based on function field F/F₂₅₆ with g(F) = 1 and N(F) ≥ 289, using
      - sharp bound on the number of rational points on curves with genus 1 [i]

There is no (27, 43, 744289)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 5986 337261 410233 965401 241867 952551 769649 605585 765556 > 16⁴³ [i]