MinT - Best Known (126−26, 126, s)-Nets in Base 3

Best Known (126−26, 126, s)-Nets in Base 3

(126−26, 126, 640)-Net over F₃ — Constructive and digital

Digital (100, 126, 640)-net over F₃, using

3² times duplication [i] based on digital (98, 124, 640)-net over F₃, using
- trace code for nets [i] based on digital (5, 31, 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]

(126−26, 126, 1475)-Net over F₃ — Digital

Digital (100, 126, 1475)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹²⁶, 1475, F₃, 26) (dual of [1475, 1349, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹²⁶, 2214, F₃, 26) (dual of [2214, 2088, 27]-code), using
  - constructionÂ X applied to C_e(25) âŠ‚ C_e(21) [i] based on
    1. linear OA(3¹²⁰, 2187, F₃, 26) (dual of [2187, 2067, 27]-code), using an extension C_e(25) of the primitive narrow-sense BCH-code C(I) with length 2186Â = 3⁷âˆ’1, defining interval IÂ =Â [1,25], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 26 [i]
    2. linear OA(3⁹⁹, 2187, F₃, 22) (dual of [2187, 2088, 23]-code), using an extension C_e(21) of the primitive narrow-sense BCH-code C(I) with length 2186Â = 3⁷âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]
    3. linear OA(3⁶, 27, F₃, 3) (dual of [27, 21, 4]-code or 27-cap in PG(5,3)), using
      - discarding factors / shortening the dual code based on linear OA(3⁶, 48, F₃, 3) (dual of [48, 42, 4]-code or 48-cap in PG(5,3)), using
        large caps in PG(5,Â b) [i]

(126−26, 126, 119326)-Net in Base 3 — Upper bound on s

There is no (100, 126, 119327)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1 310099 806360 063617 189069 971424 012589 903269 611138 755786 515239 > 3¹²⁶ [i]