Best Known (19, 19+5, s)-Nets in Base 3
(19, 19+5, 1100)-Net over F3 — Constructive and digital
Digital (19, 24, 1100)-net over F3, using
- net defined by OOA [i] based on linear OOA(324, 1100, F3, 5, 5) (dual of [(1100, 5), 5476, 6]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(324, 1101, F3, 2, 5) (dual of [(1101, 2), 2178, 6]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(32, 4, F3, 2, 2) (dual of [(4, 2), 6, 3]-NRT-code), using
- extended Reed–Solomon NRT-code RSe(2;6,3) [i]
- linear OOA(322, 1097, F3, 2, 5) (dual of [(1097, 2), 2172, 6]-NRT-code), using
- OOA 2-folding [i] based on linear OA(322, 2194, F3, 5) (dual of [2194, 2172, 6]-code), using
- construction X applied to Ce(4) ⊂ Ce(3) [i] based on
- linear OA(322, 2187, F3, 5) (dual of [2187, 2165, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(315, 2187, F3, 4) (dual of [2187, 2172, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(30, 7, F3, 0) (dual of [7, 7, 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
- construction X applied to Ce(4) ⊂ Ce(3) [i] based on
- OOA 2-folding [i] based on linear OA(322, 2194, F3, 5) (dual of [2194, 2172, 6]-code), using
- linear OOA(32, 4, F3, 2, 2) (dual of [(4, 2), 6, 3]-NRT-code), using
- (u, u+v)-construction [i] based on
- OOA stacking with additional row [i] based on linear OOA(324, 1101, F3, 2, 5) (dual of [(1101, 2), 2178, 6]-NRT-code), using
(19, 19+5, 2204)-Net over F3 — Digital
Digital (19, 24, 2204)-net over F3, using
- net defined by OOA [i] based on linear OOA(324, 2204, F3, 5, 5) (dual of [(2204, 5), 10996, 6]-NRT-code), using
- appending kth column [i] based on linear OOA(324, 2204, F3, 4, 5) (dual of [(2204, 4), 8792, 6]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(324, 2204, F3, 5) (dual of [2204, 2180, 6]-code), using
- 8 step Varšamov–Edel lengthening with (ri) = (1, 0, 1, 5 times 0) [i] based on linear OA(322, 2194, F3, 5) (dual of [2194, 2172, 6]-code), using
- construction X applied to Ce(4) ⊂ Ce(3) [i] based on
- linear OA(322, 2187, F3, 5) (dual of [2187, 2165, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(315, 2187, F3, 4) (dual of [2187, 2172, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(30, 7, F3, 0) (dual of [7, 7, 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
- construction X applied to Ce(4) ⊂ Ce(3) [i] based on
- 8 step Varšamov–Edel lengthening with (ri) = (1, 0, 1, 5 times 0) [i] based on linear OA(322, 2194, F3, 5) (dual of [2194, 2172, 6]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(324, 2204, F3, 5) (dual of [2204, 2180, 6]-code), using
- appending kth column [i] based on linear OOA(324, 2204, F3, 4, 5) (dual of [(2204, 4), 8792, 6]-NRT-code), using
(19, 19+5, 216958)-Net in Base 3 — Upper bound on s
There is no (19, 24, 216959)-net in base 3, because
- 1 times m-reduction [i] would yield (19, 23, 216959)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 94143 717117 > 323 [i]