MinT - Best Known (122, 122+29, s)-Nets in Base 3

Best Known (122, 122+29, s)-Nets in Base 3

(122, 122+29, 688)-Net over F₃ — Constructive and digital

Digital (122, 151, 688)-net over F₃, using

t-expansion [i] based on digital (121, 151, 688)-net over F₃, using
- 1 times m-reduction [i] based on digital (121, 152, 688)-net over F₃, using
  - trace code for nets [i] based on digital (7, 38, 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]

(122, 122+29, 2330)-Net over F₃ — Digital

Digital (122, 151, 2330)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹⁵¹, 2330, F₃, 29) (dual of [2330, 2179, 30]-code), using
- 119 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (4, 2, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 5 times 0, 1, 7 times 0, 1, 9 times 0, 1, 12 times 0, 1, 17 times 0, 1, 22 times 0, 1, 28 times 0) [i] based on linear OA(3¹³⁴, 2194, F₃, 29) (dual of [2194, 2060, 30]-code), using
  - constructionÂ X applied to C_e(28) âŠ‚ C_e(27) [i] based on
    1. linear OA(3¹³⁴, 2187, F₃, 29) (dual of [2187, 2053, 30]-code), using an extension C_e(28) of the primitive narrow-sense BCH-code C(I) with length 2186Â = 3⁷âˆ’1, defining interval IÂ =Â [1,28], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 29 [i]
    2. linear OA(3¹²⁷, 2187, F₃, 28) (dual of [2187, 2060, 29]-code), using an extension C_e(27) of the primitive narrow-sense BCH-code C(I) with length 2186Â = 3⁷âˆ’1, defining interval IÂ =Â [1,27], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 28 [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]

(122, 122+29, 391224)-Net in Base 3 — Upper bound on s

There is no (122, 151, 391225)-net in base 3, because

1 times m-reduction [i] would yield (122, 150, 391225)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 369993 229044 783340 710440 642604 875321 748665 161279 187331 610145 099894 030121 > 3¹⁵⁰ [i]