Best Known (161−18, 161, s)-Nets in Base 3
(161−18, 161, 177150)-Net over F3 — Constructive and digital
Digital (143, 161, 177150)-net over F3, using
- t-expansion [i] based on digital (142, 161, 177150)-net over F3, using
- net defined by OOA [i] based on linear OOA(3161, 177150, F3, 19, 19) (dual of [(177150, 19), 3365689, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(3161, 1594351, F3, 19) (dual of [1594351, 1594190, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(3161, 1594353, F3, 19) (dual of [1594353, 1594192, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(15) [i] based on
- linear OA(3157, 1594323, F3, 19) (dual of [1594323, 1594166, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [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(34, 30, F3, 2) (dual of [30, 26, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- Hamming code H(4,3) [i]
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- construction X applied to Ce(18) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(3161, 1594353, F3, 19) (dual of [1594353, 1594192, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(3161, 1594351, F3, 19) (dual of [1594351, 1594190, 20]-code), using
- net defined by OOA [i] based on linear OOA(3161, 177150, F3, 19, 19) (dual of [(177150, 19), 3365689, 20]-NRT-code), using
(161−18, 161, 531451)-Net over F3 — Digital
Digital (143, 161, 531451)-net over F3, using
- 31 times duplication [i] based on digital (142, 160, 531451)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3160, 531451, F3, 3, 18) (dual of [(531451, 3), 1594193, 19]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3160, 1594353, F3, 18) (dual of [1594353, 1594193, 19]-code), using
- 2 times code embedding in larger space [i] based on linear OA(3158, 1594351, F3, 18) (dual of [1594351, 1594193, 19]-code), using
- construction X4 applied to Ce(18) ⊂ Ce(15) [i] based on
- linear OA(3157, 1594323, F3, 19) (dual of [1594323, 1594166, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [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(327, 28, F3, 27) (dual of [28, 1, 28]-code or 28-arc in PG(26,3)), using
- dual of repetition code with length 28 [i]
- linear OA(31, 28, F3, 1) (dual of [28, 27, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X4 applied to Ce(18) ⊂ Ce(15) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(3158, 1594351, F3, 18) (dual of [1594351, 1594193, 19]-code), using
- OOA 3-folding [i] based on linear OA(3160, 1594353, F3, 18) (dual of [1594353, 1594193, 19]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3160, 531451, F3, 3, 18) (dual of [(531451, 3), 1594193, 19]-NRT-code), using
(161−18, 161, large)-Net in Base 3 — Upper bound on s
There is no (143, 161, large)-net in base 3, because
- 16 times m-reduction [i] would yield (143, 145, large)-net in base 3, but