MinT - Best Known (142−30, 142, s)-Nets in Base 3

Best Known (142−30, 142, s)-Nets in Base 3

(142−30, 142, 640)-Net over F₃ — Constructive and digital

Digital (112, 142, 640)-net over F₃, using

3² times duplication [i] based on digital (110, 140, 640)-net over F₃, using
- trace code for nets [i] based on digital (5, 35, 160)-net over F₈₁, using
  - net from sequence [i] based on digital (5, 159)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 5 and N(F) ≥ 160, using
      - a function field by SÃ©mirat [i]

(142−30, 142, 1402)-Net over F₃ — Digital

Digital (112, 142, 1402)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹⁴², 1402, F₃, 30) (dual of [1402, 1260, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹⁴², 2202, F₃, 30) (dual of [2202, 2060, 31]-code), using
  - constructionÂ X applied to C_e(30) âŠ‚ C_e(27) [i] based on
    1. linear OA(3¹⁴¹, 2187, F₃, 31) (dual of [2187, 2046, 32]-code), using an extension C_e(30) of the primitive narrow-sense BCH-code C(I) with length 2186Â = 3⁷âˆ’1, defining interval IÂ =Â [1,30], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 31 [i]
    2. linear OA(3¹²⁷, 2187, F₃, 28) (dual of [2187, 2060, 29]-code), using an extension C_e(27) of the primitive narrow-sense BCH-code C(I) with length 2186Â = 3⁷âˆ’1, defining interval IÂ =Â [1,27], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 28 [i]
    3. linear OA(3¹, 15, F₃, 1) (dual of [15, 14, 2]-code), using
      - discarding factors / shortening the dual code based on linear OA(3¹, s, F₃, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(142−30, 142, 105541)-Net in Base 3 — Upper bound on s

There is no (112, 142, 105542)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 56 392973 753933 568780 689489 986924 798717 987706 573000 049211 915447 189705 > 3¹⁴² [i]