MinT - Best Known (196−37, 196, s)-Nets in Base 3

Best Known (196−37, 196, s)-Nets in Base 3

(196−37, 196, 696)-Net over F₃ — Constructive and digital

Digital (159, 196, 696)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (2, 20, 8)-net over F₃, using
  - net from sequence [i] based on digital (2, 7)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 2 and N(F) ≥ 8, 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]

(196−37, 196, 3287)-Net over F₃ — Digital

Digital (159, 196, 3287)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3¹⁹⁶, 3287, F₃, 2, 37) (dual of [(3287, 2), 6378, 38]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3¹⁹⁶, 6574, F₃, 37) (dual of [6574, 6378, 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(3³, 13, F₃, 2) (dual of [13, 10, 3]-code), using
      - Hamming code H(3,3) [i]

(196−37, 196, 557057)-Net in Base 3 — Upper bound on s

There is no (159, 196, 557058)-net in base 3, because

1 times m-reduction [i] would yield (159, 195, 557058)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1093 080019 676025 605618 180935 831604 077583 556749 139521 258395 181402 781295 776667 471127 538743 992501 > 3¹⁹⁵ [i]