MinT - Best Known (142−29, 142, s)-Nets in Base 3

Best Known (142−29, 142, s)-Nets in Base 3

(142−29, 142, 640)-Net over F₃ — Constructive and digital

Digital (113, 142, 640)-net over F₃, using

2 times m-reduction [i] based on digital (113, 144, 640)-net over F₃, using
- trace code for nets [i] based on digital (5, 36, 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]

(142−29, 142, 1669)-Net over F₃ — Digital

Digital (113, 142, 1669)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹⁴², 1669, F₃, 29) (dual of [1669, 1527, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹⁴², 2203, F₃, 29) (dual of [2203, 2061, 30]-code), using
  - constructionÂ X applied to C([0,15]) âŠ‚ C([0,13]) [i] based on
    1. linear OA(3¹⁴¹, 2188, F₃, 31) (dual of [2188, 2047, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188Â | 3¹⁴âˆ’1, defining interval IÂ =Â [0,15], and minimum distance dÂ â‰¥ |{−15,−14,…,15}|+1Â =Â 32 (BCH-bound) [i]
    2. linear OA(3¹²⁷, 2188, F₃, 27) (dual of [2188, 2061, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188Â | 3¹⁴âˆ’1, defining interval IÂ =Â [0,13], and minimum distance dÂ â‰¥ |{−13,−12,…,13}|+1Â =Â 28 (BCH-bound) [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]

(142−29, 142, 193058)-Net in Base 3 — Upper bound on s

There is no (113, 142, 193059)-net in base 3, because

1 times m-reduction [i] would yield (113, 141, 193059)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 18 797385 839098 520235 073828 824559 118734 299498 926844 236568 857224 511373 > 3¹⁴¹ [i]