Best Known (172, 193, s)-Nets in Base 3
(172, 193, 159439)-Net over F3 — Constructive and digital
Digital (172, 193, 159439)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (1, 11, 7)-net over F3, using
- net from sequence [i] based on digital (1, 6)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 1 and N(F) ≥ 7, using
- net from sequence [i] based on digital (1, 6)-sequence over F3, using
- digital (161, 182, 159432)-net over F3, using
- net defined by OOA [i] based on linear OOA(3182, 159432, F3, 21, 21) (dual of [(159432, 21), 3347890, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(3182, 1594321, F3, 21) (dual of [1594321, 1594139, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(3182, 1594322, F3, 21) (dual of [1594322, 1594140, 22]-code), using
- 1 times truncation [i] based on linear OA(3183, 1594323, F3, 22) (dual of [1594323, 1594140, 23]-code), using
- an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- 1 times truncation [i] based on linear OA(3183, 1594323, F3, 22) (dual of [1594323, 1594140, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(3182, 1594322, F3, 21) (dual of [1594322, 1594140, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(3182, 1594321, F3, 21) (dual of [1594321, 1594139, 22]-code), using
- net defined by OOA [i] based on linear OOA(3182, 159432, F3, 21, 21) (dual of [(159432, 21), 3347890, 22]-NRT-code), using
- digital (1, 11, 7)-net over F3, using
(172, 193, 531461)-Net over F3 — Digital
Digital (172, 193, 531461)-net over F3, using
- 31 times duplication [i] based on digital (171, 192, 531461)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3192, 531461, F3, 3, 21) (dual of [(531461, 3), 1594191, 22]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3192, 1594383, F3, 21) (dual of [1594383, 1594191, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(3192, 1594384, F3, 21) (dual of [1594384, 1594192, 22]-code), using
- construction X applied to Ce(21) ⊂ Ce(15) [i] based on
- linear OA(3183, 1594323, F3, 22) (dual of [1594323, 1594140, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(3131, 1594323, F3, 16) (dual of [1594323, 1594192, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(39, 61, F3, 4) (dual of [61, 52, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(39, 80, F3, 4) (dual of [80, 71, 5]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 80 = 34−1, defining interval I = [0,2], and designed minimum distance d ≥ |I|+1 = 5 [i]
- discarding factors / shortening the dual code based on linear OA(39, 80, F3, 4) (dual of [80, 71, 5]-code), using
- construction X applied to Ce(21) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(3192, 1594384, F3, 21) (dual of [1594384, 1594192, 22]-code), using
- OOA 3-folding [i] based on linear OA(3192, 1594383, F3, 21) (dual of [1594383, 1594191, 22]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3192, 531461, F3, 3, 21) (dual of [(531461, 3), 1594191, 22]-NRT-code), using
(172, 193, large)-Net in Base 3 — Upper bound on s
There is no (172, 193, large)-net in base 3, because
- 19 times m-reduction [i] would yield (172, 174, large)-net in base 3, but