MinT - Best Known (12, 28, s)-Nets in Base 81

Best Known (12, 28, s)-Nets in Base 81

(12, 28, 232)-Net over F₈₁ — Constructive and digital

Digital (12, 28, 232)-net over F₈₁, using

(u,Â u+v)-construction [i] based on
1. digital (2, 10, 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]
2. digital (2, 18, 116)-net over F₈₁, using
  - net from sequence [i] based on digital (2, 115)-sequence over F₈₁ (see above)

(12, 28, 327)-Net over F₈₁ — Digital

Digital (12, 28, 327)-net over F₈₁, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(81²⁸, 327, F₈₁, 16) (dual of [327, 299, 17]-code), using
- 1 times truncation [i] based on linear OA(81²⁹, 328, F₈₁, 17) (dual of [328, 299, 18]-code), using
  - the BCH-code C(I) with length 328Â | 81²âˆ’1, defining interval IÂ =Â {1,6,11,…,81}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 18 [i]

(12, 28, 225056)-Net in Base 81 — Upper bound on s

There is no (12, 28, 225057)-net in base 81, because

the generalized Rao bound for nets shows that 81^mÂ â‰¥ 273901 966953 989113 934875 394314 004094 559792 603795 902081 > 81²⁸ [i]