MinT - Best Known (152−28, 152, s)-Nets in Base 3

Best Known (152−28, 152, s)-Nets in Base 3

(152−28, 152, 688)-Net over F₃ — Constructive and digital

Digital (124, 152, 688)-net over F₃, using

4 times m-reduction [i] based on digital (124, 156, 688)-net over F₃, using
- trace code for nets [i] based on digital (7, 39, 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]

(152−28, 152, 3294)-Net over F₃ — Digital

Digital (124, 152, 3294)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3¹⁵², 3294, F₃, 2, 28) (dual of [(3294, 2), 6436, 29]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3¹⁵², 6588, F₃, 28) (dual of [6588, 6436, 29]-code), using
  - constructionÂ X applied to C_e(27) âŠ‚ C_e(22) [i] based on
    1. linear OA(3¹⁴⁵, 6561, F₃, 28) (dual of [6561, 6416, 29]-code), using an extension C_e(27) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 3⁸âˆ’1, defining interval IÂ =Â [1,27], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 28 [i]
    2. linear OA(3¹²¹, 6561, F₃, 23) (dual of [6561, 6440, 24]-code), using an extension C_e(22) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 3⁸âˆ’1, defining interval IÂ =Â [1,22], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]
    3. linear OA(3⁷, 27, F₃, 4) (dual of [27, 20, 5]-code), using
      - an extension C_e(3) of the primitive narrow-sense BCH-code C(I) with length 26Â = 3³âˆ’1, defining interval IÂ =Â [1,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [i]

(152−28, 152, 457708)-Net in Base 3 — Upper bound on s

There is no (124, 152, 457709)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 3 329992 408372 651693 710598 875017 899430 967478 018747 500356 696237 377484 514033 > 3¹⁵² [i]