Best Known (199, 199+24, s)-Nets in Base 3
(199, 199+24, 132870)-Net over F3 — Constructive and digital
Digital (199, 223, 132870)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (3, 15, 10)-net over F3, using
- net from sequence [i] based on digital (3, 9)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 3 and N(F) ≥ 10, using
- net from sequence [i] based on digital (3, 9)-sequence over F3, using
- digital (184, 208, 132860)-net over F3, using
- net defined by OOA [i] based on linear OOA(3208, 132860, F3, 24, 24) (dual of [(132860, 24), 3188432, 25]-NRT-code), using
- OA 12-folding and stacking [i] based on linear OA(3208, 1594320, F3, 24) (dual of [1594320, 1594112, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(3208, 1594323, F3, 24) (dual of [1594323, 1594115, 25]-code), using
- 1 times truncation [i] based on linear OA(3209, 1594324, F3, 25) (dual of [1594324, 1594115, 26]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 1594324 | 326−1, 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(3209, 1594324, F3, 25) (dual of [1594324, 1594115, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(3208, 1594323, F3, 24) (dual of [1594323, 1594115, 25]-code), using
- OA 12-folding and stacking [i] based on linear OA(3208, 1594320, F3, 24) (dual of [1594320, 1594112, 25]-code), using
- net defined by OOA [i] based on linear OOA(3208, 132860, F3, 24, 24) (dual of [(132860, 24), 3188432, 25]-NRT-code), using
- digital (3, 15, 10)-net over F3, using
(199, 199+24, 531463)-Net over F3 — Digital
Digital (199, 223, 531463)-net over F3, using
- 31 times duplication [i] based on digital (198, 222, 531463)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3222, 531463, F3, 3, 24) (dual of [(531463, 3), 1594167, 25]-NRT-code), using
- strength reduction [i] based on linear OOA(3222, 531463, F3, 3, 25) (dual of [(531463, 3), 1594167, 26]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3222, 1594389, F3, 25) (dual of [1594389, 1594167, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,9]) [i] based on
- linear OA(3209, 1594324, F3, 25) (dual of [1594324, 1594115, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 1594324 | 326−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(3157, 1594324, F3, 19) (dual of [1594324, 1594167, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 1594324 | 326−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(313, 65, F3, 5) (dual of [65, 52, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(313, 80, F3, 5) (dual of [80, 67, 6]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 80 = 34−1, defining interval I = [0,4], and designed minimum distance d ≥ |I|+1 = 6 [i]
- discarding factors / shortening the dual code based on linear OA(313, 80, F3, 5) (dual of [80, 67, 6]-code), using
- construction X applied to C([0,12]) ⊂ C([0,9]) [i] based on
- OOA 3-folding [i] based on linear OA(3222, 1594389, F3, 25) (dual of [1594389, 1594167, 26]-code), using
- strength reduction [i] based on linear OOA(3222, 531463, F3, 3, 25) (dual of [(531463, 3), 1594167, 26]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3222, 531463, F3, 3, 24) (dual of [(531463, 3), 1594167, 25]-NRT-code), using
(199, 199+24, large)-Net in Base 3 — Upper bound on s
There is no (199, 223, large)-net in base 3, because
- 22 times m-reduction [i] would yield (199, 201, large)-net in base 3, but