MinT - Best Known (15, 15+17, s)-Nets in Base 27

Best Known (15, 15+17, s)-Nets in Base 27

(15, 15+17, 116)-Net over F₂₇ — Constructive and digital

Digital (15, 32, 116)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (3, 11, 52)-net over F₂₇, using
  - net from sequence [i] based on digital (3, 51)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 3 and N(F) ≥ 52, using
      - a function field by SÃ©mirat [i]
2. digital (4, 21, 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]

(15, 15+17, 172)-Net in Base 27 — Constructive

(15, 32, 172)-net in base 27, using

base change [i] based on digital (7, 24, 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]

(15, 15+17, 218)-Net over F₂₇ — Digital

Digital (15, 32, 218)-net over F₂₇, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(27³², 218, F₂₇, 17) (dual of [218, 186, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(27³², 364, F₂₇, 17) (dual of [364, 332, 18]-code), using
  - the expurgated narrow-sense BCH-code C(I) with length 364Â | 27²âˆ’1, defining interval IÂ =Â [0,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 18 [i]

(15, 15+17, 50958)-Net in Base 27 — Upper bound on s

There is no (15, 32, 50959)-net in base 27, because

1 times m-reduction [i] would yield (15, 31, 50959)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 235 663882 226709 034904 716689 038161 135519 923249 > 27³¹ [i]