MinT - Best Known (131, 159, s)-Nets in Base 3

Best Known (131, 159, s)-Nets in Base 3

(131, 159, 701)-Net over F₃ — Constructive and digital

Digital (131, 159, 701)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (5, 19, 13)-net over F₃, using
  - net from sequence [i] based on digital (5, 12)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₃ with g(F)Â =Â 4, N(F)Â =Â 12, and 1 place with degree 2 [i] based on function field F/F₃ with g(F) = 4 and N(F) ≥ 12, using an explicitly constructive algebraic function field [i]
2. digital (112, 140, 688)-net over F₃, using
  - trace code for nets [i] based on digital (7, 35, 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]

(131, 159, 4160)-Net over F₃ — Digital

Digital (131, 159, 4160)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹⁵⁹, 4160, F₃, 28) (dual of [4160, 4001, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹⁵⁹, 6576, F₃, 28) (dual of [6576, 6417, 29]-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OA(3¹⁴, 15, F₃, 14) (dual of [15, 1, 15]-code or 15-arc in PG(13,3)), using
      - dual of repetition code with length 15 [i]
    2. linear OA(3¹⁴⁵, 6561, F₃, 28) (dual of [6561, 6416, 29]-code), using
      - an extension C_e(27) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 3⁸âˆ’1, defining interval IÂ =Â [1,27], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 28 [i]

(131, 159, 792783)-Net in Base 3 — Upper bound on s

There is no (131, 159, 792784)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 7282 548644 129279 007991 966894 488146 667612 597404 682649 742077 631861 107747 884513 > 3¹⁵⁹ [i]