MinT - Best Known (197−37, 197, s)-Nets in Base 3

Best Known (197−37, 197, s)-Nets in Base 3

(197−37, 197, 698)-Net over F₃ — Constructive and digital

Digital (160, 197, 698)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (3, 21, 10)-net over F₃, using
  - net from sequence [i] based on digital (3, 9)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 3 and N(F) ≥ 10, using
      - an explicitly constructive algebraic function field [i]
2. digital (139, 176, 688)-net over F₃, using
  - trace code for nets [i] based on digital (7, 44, 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]

(197−37, 197, 3290)-Net over F₃ — Digital

Digital (160, 197, 3290)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3¹⁹⁷, 3290, F₃, 2, 37) (dual of [(3290, 2), 6383, 38]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3¹⁹⁷, 6580, F₃, 37) (dual of [6580, 6383, 38]-code), using
  - discarding factors / shortening the dual code based on linear OA(3¹⁹⁷, 6581, F₃, 37) (dual of [6581, 6384, 38]-code), using
    - constructionÂ X applied to C_e(36) âŠ‚ C_e(33) [i] based on
      1. linear OA(3¹⁹³, 6561, F₃, 37) (dual of [6561, 6368, 38]-code), using an extension C_e(36) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 3⁸âˆ’1, defining interval IÂ =Â [1,36], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 37 [i]
      2. linear OA(3¹⁷⁷, 6561, F₃, 34) (dual of [6561, 6384, 35]-code), using an extension C_e(33) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 3⁸âˆ’1, defining interval IÂ =Â [1,33], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 34 [i]
      3. linear OA(3⁴, 20, F₃, 2) (dual of [20, 16, 3]-code), using
        discarding factors / shortening the dual code based on linear OA(3⁴, 40, F₃, 2) (dual of [40, 36, 3]-code), using
        Hamming code H(4,3) [i]

(197−37, 197, 592117)-Net in Base 3 — Upper bound on s

There is no (160, 197, 592118)-net in base 3, because

1 times m-reduction [i] would yield (160, 196, 592118)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 3279 281349 465572 135820 556104 394543 391007 149637 695974 827709 122263 860330 063390 058215 212482 417325 > 3¹⁹⁶ [i]