Best Known (194, 221, s)-Nets in Base 3
(194, 221, 40882)-Net over F3 — Constructive and digital
Digital (194, 221, 40882)-net over F3, using
- 33 times duplication [i] based on digital (191, 218, 40882)-net over F3, using
- net defined by OOA [i] based on linear OOA(3218, 40882, F3, 27, 27) (dual of [(40882, 27), 1103596, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(3218, 531467, F3, 27) (dual of [531467, 531249, 28]-code), using
- construction X applied to C([0,13]) ⊂ C([0,12]) [i] based on
- linear OA(3217, 531442, F3, 27) (dual of [531442, 531225, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 324−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- 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(31, 25, F3, 1) (dual of [25, 24, 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 X applied to C([0,13]) ⊂ C([0,12]) [i] based on
- OOA 13-folding and stacking with additional row [i] based on linear OA(3218, 531467, F3, 27) (dual of [531467, 531249, 28]-code), using
- net defined by OOA [i] based on linear OOA(3218, 40882, F3, 27, 27) (dual of [(40882, 27), 1103596, 28]-NRT-code), using
(194, 221, 156908)-Net over F3 — Digital
Digital (194, 221, 156908)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3221, 156908, F3, 3, 27) (dual of [(156908, 3), 470503, 28]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3221, 177156, F3, 3, 27) (dual of [(177156, 3), 531247, 28]-NRT-code), using
- strength reduction [i] based on linear OOA(3221, 177156, F3, 3, 28) (dual of [(177156, 3), 531247, 29]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3221, 531468, F3, 28) (dual of [531468, 531247, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(3221, 531469, F3, 28) (dual of [531469, 531248, 29]-code), using
- construction X applied to Ce(27) ⊂ Ce(24) [i] based on
- linear OA(3217, 531441, F3, 28) (dual of [531441, 531224, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(3193, 531441, F3, 25) (dual of [531441, 531248, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(34, 28, F3, 2) (dual of [28, 24, 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(27) ⊂ Ce(24) [i] based on
- discarding factors / shortening the dual code based on linear OA(3221, 531469, F3, 28) (dual of [531469, 531248, 29]-code), using
- OOA 3-folding [i] based on linear OA(3221, 531468, F3, 28) (dual of [531468, 531247, 29]-code), using
- strength reduction [i] based on linear OOA(3221, 177156, F3, 3, 28) (dual of [(177156, 3), 531247, 29]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3221, 177156, F3, 3, 27) (dual of [(177156, 3), 531247, 28]-NRT-code), using
(194, 221, large)-Net in Base 3 — Upper bound on s
There is no (194, 221, large)-net in base 3, because
- 25 times m-reduction [i] would yield (194, 196, large)-net in base 3, but