MinT - Best Known (53−51, 53, s)-Nets in Base 25

Best Known (53−51, 53, s)-Nets in Base 25

(53−51, 53, 28)-Net over F₂₅ — Constructive and digital

Digital (2, 53, 28)-net over F₂₅, using

net from sequence [i] based on digital (2, 27)-sequence over F₂₅, using
- Niederreiter sequence [i]

(53−51, 53, 46)-Net over F₂₅ — Digital

Digital (2, 53, 46)-net over F₂₅, using

net from sequence [i] based on digital (2, 45)-sequence over F₂₅, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 2 and N(F) ≥ 46, using
  - sharp bound on the number of rational points on curves with genus 2 [i]

(53−51, 53, 77)-Net over F₂₅ — Upper bound on s (digital)

There is no digital (2, 53, 78)-net over F₂₅, because

1 times m-reduction [i] would yield digital (2, 52, 78)-net over F₂₅, but
- extracting embedded orthogonal array [i] would yield linear OA(25⁵², 78, F₂₅, 50) (dual of [78, 26, 51]-code), but
  - residual code [i] would yield OA(25², 27, S₂₅, 2), but
    - bound for OAs with strength kÂ =Â 2 [i]
    - the Rao or (dual) Hamming bound shows that MÂ â‰¥ 649 > 25² [i]

(53−51, 53, 186)-Net in Base 25 — Upper bound on s

There is no (2, 53, 187)-net in base 25, because

8 times m-reduction [i] would yield (2, 45, 187)-net in base 25, but
- extracting embedded orthogonal array [i] would yield OA(25⁴⁵, 187, S₂₅, 43), but
  - the linear programming bound shows that MÂ â‰¥ 111 488601 007381 568900 361473 791104 508011 644280 696117 448314 907960 593700 408935 546875 / 137824 248946 330259 > 25⁴⁵ [i]