MinT - Best Known (47−19, 47, s)-Nets in Base 16

Best Known (47−19, 47, s)-Nets in Base 16

Digital (28, 47, 531)-net over F₁₆, using

Digital (28, 47, 728)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16⁴⁷, 728, F₁₆, 19) (dual of [728, 681, 20]-code), using
- 81 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 1, 4 times 0, 1, 18 times 0, 1, 55 times 0) [i] based on linear OA(16⁴², 642, F₁₆, 19) (dual of [642, 600, 20]-code), using
  - trace code [i] based on linear OA(256²¹, 321, F₂₅₆, 19) (dual of [321, 300, 20]-code), using
    - extended algebraic-geometric code AG_e(F,301P) [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 (28, 47, 394500)-net in base 16, because

1 times m-reduction [i] would yield (28, 46, 394500)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 24 520365 657660 137810 116312 313132 372938 707501 595550 529376 > 16⁴⁶ [i]