MinT - Best Known (24−13, 24, s)-Nets in Base 27

Best Known (24−13, 24, s)-Nets in Base 27

(24−13, 24, 102)-Net over F₂₇ — Constructive and digital

Digital (11, 24, 102)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (1, 7, 38)-net over F₂₇, using
  - net from sequence [i] based on digital (1, 37)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 1 and N(F) ≥ 38, using
      - a function field by SÃ©mirat [i]
2. digital (4, 17, 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]

(24−13, 24, 160)-Net in Base 27 — Constructive

(11, 24, 160)-net in base 27, using

base change [i] based on digital (5, 18, 160)-net over F₈₁, using
- net from sequence [i] based on digital (5, 159)-sequence over F₈₁, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 5 and N(F) ≥ 160, using
    - a function field by SÃ©mirat [i]

(24−13, 24, 181)-Net over F₂₇ — Digital

Digital (11, 24, 181)-net over F₂₇, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(27²⁴, 181, F₂₇, 13) (dual of [181, 157, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(27²⁴, 182, F₂₇, 13) (dual of [182, 158, 14]-code), using
  - the expurgated narrow-sense BCH-code C(I) with length 182Â | 27²âˆ’1, defining interval IÂ =Â [0,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 14 [i]

(24−13, 24, 35327)-Net in Base 27 — Upper bound on s

There is no (11, 24, 35328)-net in base 27, because

1 times m-reduction [i] would yield (11, 23, 35328)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 834 494174 157228 791832 659845 839873 > 27²³ [i]