MinT - Best Known (39, 67, s)-Nets in Base 16

Best Known (39, 67, s)-Nets in Base 16

Digital (39, 67, 524)-net over F₁₆, using

1 times m-reduction [i] based on digital (39, 68, 524)-net over F₁₆, using
- trace code for nets [i] based on digital (5, 34, 262)-net over F₂₅₆, using
  - net from sequence [i] based on digital (5, 261)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

Digital (39, 67, 733)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16⁶⁷, 733, F₁₆, 28) (dual of [733, 666, 29]-code), using
- 84 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (3, 1, 0, 0, 0, 1, 9 times 0, 1, 23 times 0, 1, 44 times 0) [i] based on linear OA(16⁶⁰, 642, F₁₆, 28) (dual of [642, 582, 29]-code), using
  - trace code [i] based on linear OA(256³⁰, 321, F₂₅₆, 28) (dual of [321, 291, 29]-code), using
    - extended algebraic-geometric code AG_e(F,292P) [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 (39, 67, 233307)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 474 298506 713321 772417 507434 345176 070565 517484 759428 115981 159776 746030 569094 877396 > 16⁶⁷ [i]