MinT - Best Known (124−26, 124, s)-Nets in Base 3

Best Known (124−26, 124, s)-Nets in Base 3

(124−26, 124, 640)-Net over F₃ — Constructive and digital

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]

(124−26, 124, 1344)-Net over F₃ — Digital

Digital (98, 124, 1344)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹²⁴, 1344, F₃, 26) (dual of [1344, 1220, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹²⁴, 2205, F₃, 26) (dual of [2205, 2081, 27]-code), using
  - constructionÂ X applied to C_e(25) âŠ‚ C_e(22) [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(3¹⁰⁶, 2187, F₃, 23) (dual of [2187, 2081, 24]-code), using an extension C_e(22) of the primitive narrow-sense BCH-code C(I) with length 2186Â = 3⁷âˆ’1, defining interval IÂ =Â [1,22], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]
    3. linear OA(3⁴, 18, F₃, 2) (dual of [18, 14, 3]-code), using
      - discarding factors / shortening the dual code based on linear OA(3⁴, 40, F₃, 2) (dual of [40, 36, 3]-code), using
        Hamming code H(4,3) [i]

(124−26, 124, 100768)-Net in Base 3 — Upper bound on s

There is no (98, 124, 100769)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 145566 092585 539146 482421 591898 011115 969678 885648 814062 274835 > 3¹²⁴ [i]