MinT - Best Known (44, 77, s)-Nets in Base 16

Best Known (44, 77, s)-Nets in Base 16

Digital (44, 77, 524)-net over F₁₆, using

1 times m-reduction [i] based on digital (44, 78, 524)-net over F₁₆, using
- trace code for nets [i] based on digital (5, 39, 262)-net over F₂₅₆, using
  - net from sequence [i] based on digital (5, 261)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

Digital (44, 77, 703)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16⁷⁷, 703, F₁₆, 33) (dual of [703, 626, 34]-code), using
- 54 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (4, 0, 1, 5 times 0, 1, 14 times 0, 1, 30 times 0) [i] based on linear OA(16⁷⁰, 642, F₁₆, 33) (dual of [642, 572, 34]-code), using
  - trace code [i] based on linear OA(256³⁵, 321, F₂₅₆, 33) (dual of [321, 286, 34]-code), using
    - extended algebraic-geometric code AG_e(F,287P) [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 (44, 77, 237685)-net in base 16, because

1 times m-reduction [i] would yield (44, 76, 237685)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 32 592628 576955 264055 887863 342556 118401 305345 095188 374070 242531 709791 211742 110380 764528 873151 > 16⁷⁶ [i]