MinT - Best Known (54, 54+43, s)-Nets in Base 16

Best Known (54, 54+43, s)-Nets in Base 16

(54, 54+43, 524)-Net over F₁₆ — Constructive and digital

Digital (54, 97, 524)-net over F₁₆, using

1 times m-reduction [i] based on digital (54, 98, 524)-net over F₁₆, using
- trace code for nets [i] based on digital (5, 49, 262)-net over F₂₅₆, using
  - net from sequence [i] based on digital (5, 261)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(54, 54+43, 690)-Net over F₁₆ — Digital

Digital (54, 97, 690)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16⁹⁷, 690, F₁₆, 43) (dual of [690, 593, 44]-code), using
- discarding factors / shortening the dual code based on linear OA(16⁹⁷, 695, F₁₆, 43) (dual of [695, 598, 44]-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(16⁹⁰, 642, F₁₆, 43) (dual of [642, 552, 44]-code), using
    - trace code [i] based on linear OA(256⁴⁵, 321, F₂₅₆, 43) (dual of [321, 276, 44]-code), using
      - extended algebraic-geometric code AG_e(F,277P) [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]

(54, 54+43, 184894)-Net in Base 16 — Upper bound on s

There is no (54, 97, 184895)-net in base 16, because

1 times m-reduction [i] would yield (54, 96, 184895)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 39 406088 491650 272236 002645 333285 649227 282788 473643 295246 388754 932330 806036 200986 953781 121113 530605 209461 739464 176926 > 16⁹⁶ [i]