Best Known (181, 181+24, s)-Nets in Base 3
(181, 181+24, 44293)-Net over F3 — Constructive and digital
Digital (181, 205, 44293)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (1, 13, 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 (168, 192, 44286)-net over F3, using
- net defined by OOA [i] based on linear OOA(3192, 44286, F3, 24, 24) (dual of [(44286, 24), 1062672, 25]-NRT-code), using
- OA 12-folding and stacking [i] based on linear OA(3192, 531432, F3, 24) (dual of [531432, 531240, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(3192, 531441, F3, 24) (dual of [531441, 531249, 25]-code), using
- 1 times truncation [i] based on linear OA(3193, 531442, F3, 25) (dual of [531442, 531249, 26]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 531442 | 324−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- 1 times truncation [i] based on linear OA(3193, 531442, F3, 25) (dual of [531442, 531249, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(3192, 531441, F3, 24) (dual of [531441, 531249, 25]-code), using
- OA 12-folding and stacking [i] based on linear OA(3192, 531432, F3, 24) (dual of [531432, 531240, 25]-code), using
- net defined by OOA [i] based on linear OOA(3192, 44286, F3, 24, 24) (dual of [(44286, 24), 1062672, 25]-NRT-code), using
- digital (1, 13, 7)-net over F3, using
(181, 181+24, 177658)-Net over F3 — Digital
Digital (181, 205, 177658)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3205, 177658, F3, 2, 24) (dual of [(177658, 2), 355111, 25]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3205, 265751, F3, 2, 24) (dual of [(265751, 2), 531297, 25]-NRT-code), using
- strength reduction [i] based on linear OOA(3205, 265751, F3, 2, 25) (dual of [(265751, 2), 531297, 26]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3205, 531502, F3, 25) (dual of [531502, 531297, 26]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3204, 531501, F3, 25) (dual of [531501, 531297, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,9]) [i] based on
- linear OA(3193, 531442, F3, 25) (dual of [531442, 531249, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 324−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(3145, 531442, F3, 19) (dual of [531442, 531297, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 324−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(311, 59, F3, 5) (dual of [59, 48, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(311, 85, F3, 5) (dual of [85, 74, 6]-code), using
- construction X applied to C([0,12]) ⊂ C([0,9]) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3204, 531501, F3, 25) (dual of [531501, 531297, 26]-code), using
- OOA 2-folding [i] based on linear OA(3205, 531502, F3, 25) (dual of [531502, 531297, 26]-code), using
- strength reduction [i] based on linear OOA(3205, 265751, F3, 2, 25) (dual of [(265751, 2), 531297, 26]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3205, 265751, F3, 2, 24) (dual of [(265751, 2), 531297, 25]-NRT-code), using
(181, 181+24, large)-Net in Base 3 — Upper bound on s
There is no (181, 205, large)-net in base 3, because
- 22 times m-reduction [i] would yield (181, 183, large)-net in base 3, but