MinT - Best Known (242−52, 242, s)-Nets in Base 3

Best Known (242−52, 242, s)-Nets in Base 3

(242−52, 242, 688)-Net over F₃ — Constructive and digital

Digital (190, 242, 688)-net over F₃, using

2 times m-reduction [i] based on digital (190, 244, 688)-net over F₃, using
- trace code for nets [i] based on digital (7, 61, 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]

(242−52, 242, 1897)-Net over F₃ — Digital

Digital (190, 242, 1897)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3²⁴², 1897, F₃, 52) (dual of [1897, 1655, 53]-code), using
- discarding factors / shortening the dual code based on linear OA(3²⁴², 2200, F₃, 52) (dual of [2200, 1958, 53]-code), using
  - constructionÂ X applied to C_e(51) âŠ‚ C_e(48) [i] based on
    1. linear OA(3²³⁹, 2187, F₃, 52) (dual of [2187, 1948, 53]-code), using an extension C_e(51) of the primitive narrow-sense BCH-code C(I) with length 2186Â = 3⁷âˆ’1, defining interval IÂ =Â [1,51], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 52 [i]
    2. linear OA(3²²⁵, 2187, F₃, 49) (dual of [2187, 1962, 50]-code), using an extension C_e(48) of the primitive narrow-sense BCH-code C(I) with length 2186Â = 3⁷âˆ’1, defining interval IÂ =Â [1,48], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 49 [i]
    3. linear OA(3³, 13, F₃, 2) (dual of [13, 10, 3]-code), using
      - Hamming code H(3,3) [i]

(242−52, 242, 145574)-Net in Base 3 — Upper bound on s

There is no (190, 242, 145575)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 29 066667 408034 434062 543494 340418 778390 526249 336572 942745 345365 255708 694386 866149 908342 754358 770618 906940 552610 849437 > 3²⁴² [i]