MinT - Best Known (17, 36, s)-Nets in Base 27

Best Known (17, 36, s)-Nets in Base 27

(17, 36, 128)-Net over F₂₇ — Constructive and digital

Digital (17, 36, 128)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (4, 13, 64)-net over F₂₇, using
  - net from sequence [i] based on digital (4, 63)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 4 and N(F) ≥ 64, using
      - a function field by SÃ©mirat [i]
2. digital (4, 23, 64)-net over F₂₇, using
  - net from sequence [i] based on digital (4, 63)-sequence over F₂₇ (see above)

(17, 36, 172)-Net in Base 27 — Constructive

(17, 36, 172)-net in base 27, using

4 times m-reduction [i] based on (17, 40, 172)-net in base 27, using
- base change [i] based on digital (7, 30, 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]

(17, 36, 237)-Net over F₂₇ — Digital

Digital (17, 36, 237)-net over F₂₇, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(27³⁶, 237, F₂₇, 19) (dual of [237, 201, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(27³⁶, 364, F₂₇, 19) (dual of [364, 328, 20]-code), using
  - the expurgated narrow-sense BCH-code C(I) with length 364Â | 27²âˆ’1, defining interval IÂ =Â [0,18], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 20 [i]

(17, 36, 58770)-Net in Base 27 — Upper bound on s

There is no (17, 36, 58771)-net in base 27, because

1 times m-reduction [i] would yield (17, 35, 58771)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 125 243112 915898 276002 214517 170558 508008 460232 515551 > 27³⁵ [i]