MinT - Best Known (52, 52+41, s)-Nets in Base 16

Best Known (52, 52+41, s)-Nets in Base 16

(52, 52+41, 524)-Net over F₁₆ — Constructive and digital

Digital (52, 93, 524)-net over F₁₆, using

1 times m-reduction [i] based on digital (52, 94, 524)-net over F₁₆, using
- trace code for nets [i] based on digital (5, 47, 262)-net over F₂₅₆, using
  - net from sequence [i] based on digital (5, 261)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(52, 52+41, 692)-Net over F₁₆ — Digital

Digital (52, 93, 692)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16⁹³, 692, F₁₆, 41) (dual of [692, 599, 42]-code), using
- discarding factors / shortening the dual code based on linear OA(16⁹³, 695, F₁₆, 41) (dual of [695, 602, 42]-code), using
  - 46 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (4, 0, 1, 5 times 0, 1, 12 times 0, 1, 24 times 0) [i] based on linear OA(16⁸⁶, 642, F₁₆, 41) (dual of [642, 556, 42]-code), using
    - trace code [i] based on linear OA(256⁴³, 321, F₂₅₆, 41) (dual of [321, 278, 42]-code), using
      - extended algebraic-geometric code AG_e(F,279P) [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]

(52, 52+41, 191488)-Net in Base 16 — Upper bound on s

There is no (52, 93, 191489)-net in base 16, because

1 times m-reduction [i] would yield (52, 92, 191489)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 601 268654 483971 209450 104887 905965 747598 391838 286128 459985 707814 351722 706603 453984 118384 330861 067293 417404 088576 > 16⁹² [i]