MinT - Best Known (31−18, 31, s)-Nets in Base 81

Best Known (31−18, 31, s)-Nets in Base 81

(31−18, 31, 232)-Net over F₈₁ — Constructive and digital

Digital (13, 31, 232)-net over F₈₁, using

(u,Â u+v)-construction [i] based on
1. digital (2, 11, 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, 20, 116)-net over F₈₁, using
  - net from sequence [i] based on digital (2, 115)-sequence over F₈₁ (see above)

(31−18, 31, 315)-Net over F₈₁ — Digital

Digital (13, 31, 315)-net over F₈₁, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(81³¹, 315, F₈₁, 18) (dual of [315, 284, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(81³¹, 328, F₈₁, 18) (dual of [328, 297, 19]-code), using
  - the BCH-code C(I) with length 328Â | 81²âˆ’1, defining interval IÂ =Â {1,6,11,…,86}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [i]

(31−18, 31, 194232)-Net in Base 81 — Upper bound on s

There is no (13, 31, 194233)-net in base 81, because

the generalized Rao bound for nets shows that 81^mÂ â‰¥ 145558 231120 315004 947148 362466 623876 071454 561149 041495 342161 > 81³¹ [i]