Best Known (193, 193+24, s)-Nets in Base 3
(193, 193+24, 132864)-Net over F3 — Constructive and digital
Digital (193, 217, 132864)-net over F3, using
- 1 times m-reduction [i] based on digital (193, 218, 132864)-net over F3, using
- net defined by OOA [i] based on linear OOA(3218, 132864, F3, 25, 25) (dual of [(132864, 25), 3321382, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(3218, 1594369, F3, 25) (dual of [1594369, 1594151, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(3218, 1594371, F3, 25) (dual of [1594371, 1594153, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(19) [i] based on
- linear OA(3209, 1594323, F3, 25) (dual of [1594323, 1594114, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(3170, 1594323, F3, 20) (dual of [1594323, 1594153, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(39, 48, F3, 4) (dual of [48, 39, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(39, 80, F3, 4) (dual of [80, 71, 5]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 80 = 34−1, defining interval I = [0,2], and designed minimum distance d ≥ |I|+1 = 5 [i]
- discarding factors / shortening the dual code based on linear OA(39, 80, F3, 4) (dual of [80, 71, 5]-code), using
- construction X applied to Ce(24) ⊂ Ce(19) [i] based on
- discarding factors / shortening the dual code based on linear OA(3218, 1594371, F3, 25) (dual of [1594371, 1594153, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(3218, 1594369, F3, 25) (dual of [1594369, 1594151, 26]-code), using
- net defined by OOA [i] based on linear OOA(3218, 132864, F3, 25, 25) (dual of [(132864, 25), 3321382, 26]-NRT-code), using
(193, 193+24, 529003)-Net over F3 — Digital
Digital (193, 217, 529003)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3217, 529003, F3, 3, 24) (dual of [(529003, 3), 1586792, 25]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3217, 531456, F3, 3, 24) (dual of [(531456, 3), 1594151, 25]-NRT-code), using
- 1 step truncation [i] based on linear OOA(3218, 531457, F3, 3, 25) (dual of [(531457, 3), 1594153, 26]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3218, 1594371, F3, 25) (dual of [1594371, 1594153, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(19) [i] based on
- linear OA(3209, 1594323, F3, 25) (dual of [1594323, 1594114, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(3170, 1594323, F3, 20) (dual of [1594323, 1594153, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(39, 48, F3, 4) (dual of [48, 39, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(39, 80, F3, 4) (dual of [80, 71, 5]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 80 = 34−1, defining interval I = [0,2], and designed minimum distance d ≥ |I|+1 = 5 [i]
- discarding factors / shortening the dual code based on linear OA(39, 80, F3, 4) (dual of [80, 71, 5]-code), using
- construction X applied to Ce(24) ⊂ Ce(19) [i] based on
- OOA 3-folding [i] based on linear OA(3218, 1594371, F3, 25) (dual of [1594371, 1594153, 26]-code), using
- 1 step truncation [i] based on linear OOA(3218, 531457, F3, 3, 25) (dual of [(531457, 3), 1594153, 26]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3217, 531456, F3, 3, 24) (dual of [(531456, 3), 1594151, 25]-NRT-code), using
(193, 193+24, large)-Net in Base 3 — Upper bound on s
There is no (193, 217, large)-net in base 3, because
- 22 times m-reduction [i] would yield (193, 195, large)-net in base 3, but