MinT - Best Known (32, 32+33, s)-Nets in Base 27

Best Known (32, 32+33, s)-Nets in Base 27

(32, 32+33, 170)-Net over F₂₇ — Constructive and digital

Digital (32, 65, 170)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (7, 23, 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]
2. digital (9, 42, 88)-net over F₂₇, using
  - net from sequence [i] based on digital (9, 87)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 9 and N(F) ≥ 88, using
      - a function field by SÃ©mirat [i]

(32, 32+33, 224)-Net in Base 27 — Constructive

(32, 65, 224)-net in base 27, using

11 times m-reduction [i] based on (32, 76, 224)-net in base 27, using
- base change [i] based on digital (13, 57, 224)-net over F₈₁, using
  - net from sequence [i] based on digital (13, 223)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 13 and N(F) ≥ 224, using
      - a function field by SÃ©mirat [i]

(32, 32+33, 417)-Net over F₂₇ — Digital

Digital (32, 65, 417)-net over F₂₇, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(27⁶⁵, 417, F₂₇, 33) (dual of [417, 352, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(27⁶⁵, 728, F₂₇, 33) (dual of [728, 663, 34]-code), using
  - the primitive BCH-code C(I) with length 728Â = 27²âˆ’1, defining interval IÂ =Â {3,4,…,35}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 34 [i]

(32, 32+33, 138993)-Net in Base 27 — Upper bound on s

There is no (32, 65, 138994)-net in base 27, because

1 times m-reduction [i] would yield (32, 64, 138994)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 40 485221 629897 814790 051023 460928 461839 966735 033571 145117 244350 094308 957883 004064 478318 815905 > 27⁶⁴ [i]