MinT - Best Known (173, 173+35, s)-Nets in Base 3

Best Known (173, 173+35, s)-Nets in Base 3

(173, 173+35, 1480)-Net over F₃ — Constructive and digital

Digital (173, 208, 1480)-net over F₃, using

t-expansion [i] based on digital (172, 208, 1480)-net over F₃, using
- trace code for nets [i] based on digital (16, 52, 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]

(173, 173+35, 7508)-Net over F₃ — Digital

Digital (173, 208, 7508)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3²⁰⁸, 7508, F₃, 2, 35) (dual of [(7508, 2), 14808, 36]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3²⁰⁸, 9846, F₃, 2, 35) (dual of [(9846, 2), 19484, 36]-NRT-code), using
  - OOA 2-folding [i] based on linear OA(3²⁰⁸, 19692, F₃, 35) (dual of [19692, 19484, 36]-code), using
    - constructionÂ X applied to C_e(34) âŠ‚ C_e(33) [i] based on
      1. linear OA(3²⁰⁸, 19683, F₃, 35) (dual of [19683, 19475, 36]-code), using an extension C_e(34) of the primitive narrow-sense BCH-code C(I) with length 19682Â = 3⁹âˆ’1, defining interval IÂ =Â [1,34], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 35 [i]
      2. linear OA(3¹⁹⁹, 19683, F₃, 34) (dual of [19683, 19484, 35]-code), using an extension C_e(33) of the primitive narrow-sense BCH-code C(I) with length 19682Â = 3⁹âˆ’1, defining interval IÂ =Â [1,33], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 34 [i]
      3. linear OA(3⁰, 9, F₃, 0) (dual of [9, 9, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(3⁰, s, F₃, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(173, 173+35, 2315074)-Net in Base 3 — Upper bound on s

There is no (173, 208, 2315075)-net in base 3, because

1 times m-reduction [i] would yield (173, 207, 2315075)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 580 898343 916581 578057 771186 726365 471835 899407 651670 931701 971670 918242 239204 174020 395371 248705 367271 > 3²⁰⁷ [i]