MinT - Best Known (50−25, 50, s)-Nets in Base 27

Best Known (50−25, 50, s)-Nets in Base 27

(50−25, 50, 158)-Net over F₂₇ — Constructive and digital

Digital (25, 50, 158)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (6, 18, 76)-net over F₂₇, using
  - net from sequence [i] based on digital (6, 75)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 6 and N(F) ≥ 76, using
      - a function field by SÃ©mirat [i]
2. digital (7, 32, 82)-net over F₂₇, using
  - net from sequence [i] based on digital (7, 81)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 7 and N(F) ≥ 82, using
      - a function field by SÃ©mirat [i]

(50−25, 50, 172)-Net in Base 27 — Constructive

(25, 50, 172)-net in base 27, using

22 times m-reduction [i] based on (25, 72, 172)-net in base 27, using
- base change [i] based on digital (7, 54, 172)-net over F₈₁, using
  - net from sequence [i] based on digital (7, 171)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 7 and N(F) ≥ 172, using
      - a function field by SÃ©mirat [i]

(50−25, 50, 396)-Net over F₂₇ — Digital

Digital (25, 50, 396)-net over F₂₇, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(27⁵⁰, 396, F₂₇, 25) (dual of [396, 346, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(27⁵⁰, 728, F₂₇, 25) (dual of [728, 678, 26]-code), using
  - the primitive BCH-code C(I) with length 728Â = 27²âˆ’1, defining interval IÂ =Â {3,4,…,27}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 26 [i]

(50−25, 50, 142267)-Net in Base 27 — Upper bound on s

There is no (25, 50, 142268)-net in base 27, because

1 times m-reduction [i] would yield (25, 49, 142268)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 13704 030032 857461 816370 760161 542017 084758 908138 770292 679353 478775 886657 > 27⁴⁹ [i]