MinT - Best Known (72, 72+20, s)-Nets in Base 3

Best Known (72, 72+20, s)-Nets in Base 3

(72, 72+20, 464)-Net over F₃ — Constructive and digital

Digital (72, 92, 464)-net over F₃, using

t-expansion [i] based on digital (71, 92, 464)-net over F₃, using
- trace code for nets [i] based on digital (2, 23, 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]

(72, 72+20, 1097)-Net over F₃ — Digital

Digital (72, 92, 1097)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3⁹², 1097, F₃, 2, 20) (dual of [(1097, 2), 2102, 21]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3⁹², 2194, F₃, 20) (dual of [2194, 2102, 21]-code), using
  - constructionÂ X applied to C_e(19) âŠ‚ C_e(18) [i] based on
    1. linear OA(3⁹², 2187, F₃, 20) (dual of [2187, 2095, 21]-code), using an extension C_e(19) of the primitive narrow-sense BCH-code C(I) with length 2186Â = 3⁷âˆ’1, defining interval IÂ =Â [1,19], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 20 [i]
    2. linear OA(3⁸⁵, 2187, F₃, 19) (dual of [2187, 2102, 20]-code), using an extension C_e(18) of the primitive narrow-sense BCH-code C(I) with length 2186Â = 3⁷âˆ’1, defining interval IÂ =Â [1,18], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [i]
    3. linear OA(3⁰, 7, F₃, 0) (dual of [7, 7, 1]-code), using
      - discarding factors / shortening the dual code based on linear OA(3⁰, s, F₃, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(72, 72+20, 55512)-Net in Base 3 — Upper bound on s

There is no (72, 92, 55513)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 78 564617 085722 053617 850078 234540 879320 449817 > 3⁹² [i]