Best Known (121−11, 121, s)-Nets in Base 3
(121−11, 121, 1677837)-Net over F3 — Constructive and digital
Digital (110, 121, 1677837)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (10, 15, 117)-net over F3, using
- net defined by OOA [i] based on linear OOA(315, 117, F3, 5, 5) (dual of [(117, 5), 570, 6]-NRT-code), using
- appending kth column [i] based on linear OOA(315, 117, F3, 4, 5) (dual of [(117, 4), 453, 6]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(34, 40, F3, 4, 2) (dual of [(40, 4), 156, 3]-NRT-code), using
- appending 2 arbitrary columns [i] based on linear OOA(34, 40, F3, 2, 2) (dual of [(40, 2), 76, 3]-NRT-code), using
- appending kth column [i] based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- Hamming code H(4,3) [i]
- appending kth column [i] based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- appending 2 arbitrary columns [i] based on linear OOA(34, 40, F3, 2, 2) (dual of [(40, 2), 76, 3]-NRT-code), using
- linear OOA(311, 77, F3, 4, 5) (dual of [(77, 4), 297, 6]-NRT-code), using
- extracting embedded OOA [i] based on digital (6, 11, 77)-net over F3, using
- linear OOA(34, 40, F3, 4, 2) (dual of [(40, 4), 156, 3]-NRT-code), using
- (u, u+v)-construction [i] based on
- appending kth column [i] based on linear OOA(315, 117, F3, 4, 5) (dual of [(117, 4), 453, 6]-NRT-code), using
- net defined by OOA [i] based on linear OOA(315, 117, F3, 5, 5) (dual of [(117, 5), 570, 6]-NRT-code), using
- digital (95, 106, 1677720)-net over F3, using
- net defined by OOA [i] based on linear OOA(3106, 1677720, F3, 11, 11) (dual of [(1677720, 11), 18454814, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(3106, 8388601, F3, 11) (dual of [8388601, 8388495, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(3106, large, F3, 11) (dual of [large, large−106, 12]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 14348906 = 315−1, defining interval I = [0,10], and designed minimum distance d ≥ |I|+1 = 12 [i]
- discarding factors / shortening the dual code based on linear OA(3106, large, F3, 11) (dual of [large, large−106, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(3106, 8388601, F3, 11) (dual of [8388601, 8388495, 12]-code), using
- net defined by OOA [i] based on linear OOA(3106, 1677720, F3, 11, 11) (dual of [(1677720, 11), 18454814, 12]-NRT-code), using
- digital (10, 15, 117)-net over F3, using
(121−11, 121, 4768013)-Net over F3 — Digital
Digital (110, 121, 4768013)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3121, 4768013, F3, 11) (dual of [4768013, 4767892, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(3121, large, F3, 11) (dual of [large, large−121, 12]-code), using
- strength reduction [i] based on linear OA(3121, large, F3, 13) (dual of [large, large−121, 14]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 14348908 | 330−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- strength reduction [i] based on linear OA(3121, large, F3, 13) (dual of [large, large−121, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(3121, large, F3, 11) (dual of [large, large−121, 12]-code), using
(121−11, 121, large)-Net in Base 3 — Upper bound on s
There is no (110, 121, large)-net in base 3, because
- 9 times m-reduction [i] would yield (110, 112, large)-net in base 3, but