MinT - Best Known (152, 152+32, s)-Nets in Base 3

Best Known (152, 152+32, s)-Nets in Base 3

(152, 152+32, 896)-Net over F₃ — Constructive and digital

Digital (152, 184, 896)-net over F₃, using

t-expansion [i] based on digital (151, 184, 896)-net over F₃, using
- trace code for nets [i] based on digital (13, 46, 224)-net over F₈₁, using
  - net from sequence [i] based on digital (13, 223)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 13 and N(F) ≥ 224, using
      - a function field by SÃ©mirat [i]

(152, 152+32, 4871)-Net over F₃ — Digital

Digital (152, 184, 4871)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹⁸⁴, 4871, F₃, 32) (dual of [4871, 4687, 33]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹⁸⁴, 6616, F₃, 32) (dual of [6616, 6432, 33]-code), using
  - constructionÂ X applied to C_e(31) âŠ‚ C_e(24) [i] based on
    1. linear OA(3¹⁶⁹, 6561, F₃, 32) (dual of [6561, 6392, 33]-code), using an extension C_e(31) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 3⁸âˆ’1, defining interval IÂ =Â [1,31], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 32 [i]
    2. linear OA(3¹²⁹, 6561, F₃, 25) (dual of [6561, 6432, 26]-code), using an extension C_e(24) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 3⁸âˆ’1, defining interval IÂ =Â [1,24], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 25 [i]
    3. linear OA(3¹⁵, 55, F₃, 6) (dual of [55, 40, 7]-code), using
      - discarding factors / shortening the dual code based on linear OA(3¹⁵, 85, F₃, 6) (dual of [85, 70, 7]-code), using
        codes constructed by inverting constructionÂ Y₁ [i]

(152, 152+32, 1043269)-Net in Base 3 — Upper bound on s

There is no (152, 184, 1043270)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 6170 368183 707562 019612 205871 234953 100390 371585 644570 974612 601826 522177 990561 827544 395425 > 3¹⁸⁴ [i]