MinT - Best Known (198, 241, s)-Nets in Base 3

Best Known (198, 241, s)-Nets in Base 3

(198, 241, 1480)-Net over F₃ — Constructive and digital

Digital (198, 241, 1480)-net over F₃, using

3¹ times duplication [i] based on digital (197, 240, 1480)-net over F₃, using
- t-expansion [i] based on digital (196, 240, 1480)-net over F₃, using
  - trace code for nets [i] based on digital (16, 60, 370)-net over F₈₁, using
    - net from sequence [i] based on digital (16, 369)-sequence over F₈₁, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 16 and N(F) ≥ 370, using
        a function field by GarcÃa and Quoos [i]

(198, 241, 4970)-Net over F₃ — Digital

Digital (198, 241, 4970)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3²⁴¹, 4970, F₃, 43) (dual of [4970, 4729, 44]-code), using
- discarding factors / shortening the dual code based on linear OA(3²⁴¹, 6617, F₃, 43) (dual of [6617, 6376, 44]-code), using
  - constructionÂ X applied to C_e(42) âŠ‚ C_e(34) [i] based on
    1. linear OA(3²²⁵, 6561, F₃, 43) (dual of [6561, 6336, 44]-code), using an extension C_e(42) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 3⁸âˆ’1, defining interval IÂ =Â [1,42], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 43 [i]
    2. linear OA(3¹⁸⁵, 6561, F₃, 35) (dual of [6561, 6376, 36]-code), using an extension C_e(34) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 3⁸âˆ’1, defining interval IÂ =Â [1,34], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 35 [i]
    3. linear OA(3¹⁶, 56, F₃, 7) (dual of [56, 40, 8]-code), using
      - discarding factors / shortening the dual code based on linear OA(3¹⁶, 82, F₃, 7) (dual of [82, 66, 8]-code), using
        generator matrices provided on Edelâ€™s homepage [i]

(198, 241, 1231032)-Net in Base 3 — Upper bound on s

There is no (198, 241, 1231033)-net in base 3, because

1 times m-reduction [i] would yield (198, 240, 1231033)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 3 229248 510297 193507 886362 295084 007545 262957 525385 404117 997631 049074 915591 491438 382154 383057 619345 970213 000552 677539 > 3²⁴⁰ [i]