MinT - Best Known (14−8, 14, s)-Nets in Base 27

Best Known (14−8, 14, s)-Nets in Base 27

(14−8, 14, 84)-Net over F₂₇ — Constructive and digital

Digital (6, 14, 84)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (0, 2, 28)-net over F₂₇, using
  - kÂ =Â 2 construction [i]
2. digital (0, 4, 28)-net over F₂₇, using
  - net from sequence [i] based on digital (0, 27)-sequence over F₂₇, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 0 and N(F) ≥ 28, using
      - the rational function field F₂₇(x) [i]
    - Niederreiter sequence [i]
3. digital (0, 8, 28)-net over F₂₇, using
  - net from sequence [i] based on digital (0, 27)-sequence over F₂₇ (see above)

(14−8, 14, 116)-Net in Base 27 — Constructive

(6, 14, 116)-net in base 27, using

2 times m-reduction [i] based on (6, 16, 116)-net in base 27, using
- base change [i] based on digital (2, 12, 116)-net over F₈₁, using
  - net from sequence [i] based on digital (2, 115)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 2 and N(F) ≥ 116, using
      - a function field by SÃ©mirat [i]

(14−8, 14, 143)-Net over F₂₇ — Digital

Digital (6, 14, 143)-net over F₂₇, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(27¹⁴, 143, F₂₇, 8) (dual of [143, 129, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(27¹⁴, 182, F₂₇, 8) (dual of [182, 168, 9]-code), using
  - the expurgated narrow-sense BCH-code C(I) with length 182Â | 27²âˆ’1, defining interval IÂ =Â [0,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]

(14−8, 14, 8705)-Net in Base 27 — Upper bound on s

There is no (6, 14, 8706)-net in base 27, because

the generalized Rao bound for nets shows that 27^mÂ â‰¥ 109 467892 108799 580201 > 27¹⁴ [i]