MinT - Best Known (167−20, 167, s)-Nets in Base 3

Best Known (167−20, 167, s)-Nets in Base 3

(167−20, 167, 53149)-Net over F₃ — Constructive and digital

Digital (147, 167, 53149)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (0, 10, 4)-net over F₃, using
  - net from sequence [i] based on digital (0, 3)-sequence over F₃, using
    - Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 0 and N(F) ≥ 4, using
      - the rational function field F₃(x) [i]
    - Niederreiter sequence [i]
2. digital (137, 157, 53145)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3¹⁵⁷, 53145, F₃, 20, 20) (dual of [(53145, 20), 1062743, 21]-NRT-code), using
    - OA 10-folding and stacking [i] based on linear OA(3¹⁵⁷, 531450, F₃, 20) (dual of [531450, 531293, 21]-code), using
      - discarding factors / shortening the dual code based on linear OA(3¹⁵⁷, 531453, F₃, 20) (dual of [531453, 531296, 21]-code), using
        constructionÂ X applied to C_e(19) âŠ‚ C_e(18) [i] based on
        linear OA(3¹⁵⁷, 531441, F₃, 20) (dual of [531441, 531284, 21]-code), using an extension C_e(19) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 3¹²âˆ’1, defining interval IÂ =Â [1,19], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 20 [i]
        linear OA(3¹⁴⁵, 531441, F₃, 19) (dual of [531441, 531296, 20]-code), using an extension C_e(18) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 3¹²âˆ’1, defining interval IÂ =Â [1,18], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [i]
        linear OA(3⁰, 12, F₃, 0) (dual of [12, 12, 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]

(167−20, 167, 177162)-Net over F₃ — Digital

Digital (147, 167, 177162)-net over F₃, using

3¹ times duplication [i] based on digital (146, 166, 177162)-net over F₃, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3¹⁶⁶, 177162, F₃, 3, 20) (dual of [(177162, 3), 531320, 21]-NRT-code), using
  - 1 times NRT-code embedding in larger space [i] based on linear OOA(3¹⁶³, 177161, F₃, 3, 20) (dual of [(177161, 3), 531320, 21]-NRT-code), using
    - OOA 3-folding [i] based on linear OA(3¹⁶³, 531483, F₃, 20) (dual of [531483, 531320, 21]-code), using
      - constructionÂ X applied to C_e(19) âŠ‚ C_e(15) [i] based on
        linear OA(3¹⁵⁷, 531441, F₃, 20) (dual of [531441, 531284, 21]-code), using an extension C_e(19) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 3¹²âˆ’1, defining interval IÂ =Â [1,19], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 20 [i]
        linear OA(3¹²¹, 531441, F₃, 16) (dual of [531441, 531320, 17]-code), using an extension C_e(15) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 3¹²âˆ’1, defining interval IÂ =Â [1,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]
        linear OA(3⁶, 42, F₃, 3) (dual of [42, 36, 4]-code or 42-cap in PG(5,3)), using
        discarding factors / shortening the dual code based on linear OA(3⁶, 48, F₃, 3) (dual of [48, 42, 4]-code or 48-cap in PG(5,3)), using
        large caps in PG(5,Â b) [i]

(167−20, 167, large)-Net in Base 3 — Upper bound on s

There is no (147, 167, large)-net in base 3, because

18 times m-reduction [i] would yield (147, 149, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]

Best Known (167−20, 167, s)-Nets in Base 3

(167−20, 167, 53149)-Net over F3 — Constructive and digital

(167−20, 167, 177162)-Net over F3 — Digital

(167−20, 167, large)-Net in Base 3 — Upper bound on s

(167−20, 167, 53149)-Net over F₃ — Constructive and digital

(167−20, 167, 177162)-Net over F₃ — Digital