MinT - Best Known (208−41, 208, s)-Nets in Base 3

Best Known (208−41, 208, s)-Nets in Base 3

(208−41, 208, 688)-Net over F₃ — Constructive and digital

Digital (167, 208, 688)-net over F₃, using

t-expansion [i] based on digital (166, 208, 688)-net over F₃, using
- 4 times m-reduction [i] based on digital (166, 212, 688)-net over F₃, using
  - trace code for nets [i] based on digital (7, 53, 172)-net over F₈₁, using
    - net from sequence [i] based on digital (7, 171)-sequence over F₈₁, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 7 and N(F) ≥ 172, using
        a function field by SÃ©mirat [i]

(208−41, 208, 2437)-Net over F₃ — Digital

Digital (167, 208, 2437)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3²⁰⁸, 2437, F₃, 41) (dual of [2437, 2229, 42]-code), using
- 225 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (4, 1, 1, 1, 0, 1, 0, 0, 0, 1, 4 times 0, 1, 6 times 0, 1, 8 times 0, 1, 11 times 0, 1, 15 times 0, 1, 20 times 0, 1, 26 times 0, 1, 32 times 0, 1, 38 times 0, 1, 46 times 0) [i] based on linear OA(3¹⁹⁰, 2194, F₃, 41) (dual of [2194, 2004, 42]-code), using
  - constructionÂ X applied to C_e(40) âŠ‚ C_e(39) [i] based on
    1. linear OA(3¹⁹⁰, 2187, F₃, 41) (dual of [2187, 1997, 42]-code), using an extension C_e(40) of the primitive narrow-sense BCH-code C(I) with length 2186Â = 3⁷âˆ’1, defining interval IÂ =Â [1,40], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 41 [i]
    2. linear OA(3¹⁸³, 2187, F₃, 40) (dual of [2187, 2004, 41]-code), using an extension C_e(39) of the primitive narrow-sense BCH-code C(I) with length 2186Â = 3⁷âˆ’1, defining interval IÂ =Â [1,39], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 40 [i]
    3. linear OA(3⁰, 7, F₃, 0) (dual of [7, 7, 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]

(208−41, 208, 360128)-Net in Base 3 — Upper bound on s

There is no (167, 208, 360129)-net in base 3, because

1 times m-reduction [i] would yield (167, 207, 360129)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 580 908179 730437 338284 828247 071006 493749 113746 759788 981322 344470 789109 636845 963085 025298 902831 698321 > 3²⁰⁷ [i]