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

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

(27, 27+27, 164)-Net over F₂₇ — Constructive and digital

Digital (27, 54, 164)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (7, 20, 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 (7, 34, 82)-net over F₂₇, using
  - net from sequence [i] based on digital (7, 81)-sequence over F₂₇ (see above)

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

(27, 54, 224)-net in base 27, using

2 times m-reduction [i] based on (27, 56, 224)-net in base 27, using
- base change [i] based on digital (13, 42, 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]

(27, 27+27, 413)-Net over F₂₇ — Digital

Digital (27, 54, 413)-net over F₂₇, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(27⁵⁴, 413, F₂₇, 27) (dual of [413, 359, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(27⁵⁴, 735, F₂₇, 27) (dual of [735, 681, 28]-code), using
  - constructionÂ X applied to C([0,13]) âŠ‚ C([0,12]) [i] based on
    1. linear OA(27⁵³, 730, F₂₇, 27) (dual of [730, 677, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 730Â | 27⁴âˆ’1, defining interval IÂ =Â [0,13], and minimum distance dÂ â‰¥ |{−13,−12,…,13}|+1Â =Â 28 (BCH-bound) [i]
    2. linear OA(27⁴⁹, 730, F₂₇, 25) (dual of [730, 681, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 730Â | 27⁴âˆ’1, defining interval IÂ =Â [0,12], and minimum distance dÂ â‰¥ |{−12,−11,…,12}|+1Â =Â 26 (BCH-bound) [i]
    3. linear OA(27¹, 5, F₂₇, 1) (dual of [5, 4, 2]-code), using
      - discarding factors / shortening the dual code based on linear OA(27¹, s, F₂₇, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(27, 27+27, 149270)-Net in Base 27 — Upper bound on s

There is no (27, 54, 149271)-net in base 27, because

1 times m-reduction [i] would yield (27, 53, 149271)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 7282 823688 096223 645211 112513 162601 243853 663220 418102 277527 619121 487029 930463 > 27⁵³ [i]