Best Known (193, 193+27, s)-Nets in Base 3
(193, 193+27, 40882)-Net over F3 — Constructive and digital
Digital (193, 220, 40882)-net over F3, using
- 32 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
(193, 193+27, 149588)-Net over F3 — Digital
Digital (193, 220, 149588)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3220, 149588, F3, 3, 27) (dual of [(149588, 3), 448544, 28]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3220, 177156, F3, 3, 27) (dual of [(177156, 3), 531248, 28]-NRT-code), using
- 31 times duplication [i] based on linear OOA(3219, 177156, F3, 3, 27) (dual of [(177156, 3), 531249, 28]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3219, 531468, F3, 27) (dual of [531468, 531249, 28]-code), using
- 1 times code embedding in larger space [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
- 1 times code embedding in larger space [i] based on linear OA(3218, 531467, F3, 27) (dual of [531467, 531249, 28]-code), using
- OOA 3-folding [i] based on linear OA(3219, 531468, F3, 27) (dual of [531468, 531249, 28]-code), using
- 31 times duplication [i] based on linear OOA(3219, 177156, F3, 3, 27) (dual of [(177156, 3), 531249, 28]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3220, 177156, F3, 3, 27) (dual of [(177156, 3), 531248, 28]-NRT-code), using
(193, 193+27, large)-Net in Base 3 — Upper bound on s
There is no (193, 220, large)-net in base 3, because
- 25 times m-reduction [i] would yield (193, 195, large)-net in base 3, but