MinT - Best Known (174−33, 174, s)-Nets in Base 4

Best Known (174−33, 174, s)-Nets in Base 4

(174−33, 174, 1126)-Net over F₄ — Constructive and digital

Digital (141, 174, 1126)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (26, 42, 98)-net over F₄, using
  - trace code for nets [i] based on digital (5, 21, 49)-net over F₁₆, using
    - net from sequence [i] based on digital (5, 48)-sequence over F₁₆, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 5 and N(F) ≥ 49, using
        a function field by SÃ©mirat [i]
2. digital (99, 132, 1028)-net over F₄, using
  - trace code for nets [i] based on digital (0, 33, 257)-net over F₂₅₆, using
    - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
      - Niederreiter sequence [i]

(174−33, 174, 9455)-Net over F₄ — Digital

Digital (141, 174, 9455)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹⁷⁴, 9455, F₄, 33) (dual of [9455, 9281, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(4¹⁷⁴, 16404, F₄, 33) (dual of [16404, 16230, 34]-code), using
  - constructionÂ X applied to C([0,16]) âŠ‚ C([0,14]) [i] based on
    1. linear OA(4¹⁶⁹, 16385, F₄, 33) (dual of [16385, 16216, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385Â | 4¹⁴âˆ’1, defining interval IÂ =Â [0,16], and minimum distance dÂ â‰¥ |{−16,−15,…,16}|+1Â =Â 34 (BCH-bound) [i]
    2. linear OA(4¹⁵⁵, 16385, F₄, 29) (dual of [16385, 16230, 30]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385Â | 4¹⁴âˆ’1, defining interval IÂ =Â [0,14], and minimum distance dÂ â‰¥ |{−14,−13,…,14}|+1Â =Â 30 (BCH-bound) [i]
    3. linear OA(4⁵, 19, F₄, 3) (dual of [19, 14, 4]-code or 19-cap in PG(4,4)), using
      - discarding factors / shortening the dual code based on linear OA(4⁵, 41, F₄, 3) (dual of [41, 36, 4]-code or 41-cap in PG(4,4)), using
        a cap found by Tallinii [i]

(174−33, 174, 7331458)-Net in Base 4 — Upper bound on s

There is no (141, 174, 7331459)-net in base 4, because

1 times m-reduction [i] would yield (141, 173, 7331459)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 143 343897 322655 586413 967561 815026 256061 947927 788797 015224 033620 947012 072538 562862 386329 100877 966289 881304 > 4¹⁷³ [i]