Best Known (219, 219+28, s)-Nets in Base 3
(219, 219+28, 113884)-Net over F3 — Constructive and digital
Digital (219, 247, 113884)-net over F3, using
- 31 times duplication [i] based on digital (218, 246, 113884)-net over F3, using
- net defined by OOA [i] based on linear OOA(3246, 113884, F3, 28, 28) (dual of [(113884, 28), 3188506, 29]-NRT-code), using
- OA 14-folding and stacking [i] based on linear OA(3246, 1594376, F3, 28) (dual of [1594376, 1594130, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(3246, 1594386, F3, 28) (dual of [1594386, 1594140, 29]-code), using
- construction X applied to Ce(27) ⊂ Ce(21) [i] based on
- linear OA(3235, 1594323, F3, 28) (dual of [1594323, 1594088, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(3183, 1594323, F3, 22) (dual of [1594323, 1594140, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(311, 63, F3, 5) (dual of [63, 52, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(311, 85, F3, 5) (dual of [85, 74, 6]-code), using
- construction X applied to Ce(27) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(3246, 1594386, F3, 28) (dual of [1594386, 1594140, 29]-code), using
- OA 14-folding and stacking [i] based on linear OA(3246, 1594376, F3, 28) (dual of [1594376, 1594130, 29]-code), using
- net defined by OOA [i] based on linear OOA(3246, 113884, F3, 28, 28) (dual of [(113884, 28), 3188506, 29]-NRT-code), using
(219, 219+28, 398596)-Net over F3 — Digital
Digital (219, 247, 398596)-net over F3, using
- 31 times duplication [i] based on digital (218, 246, 398596)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3246, 398596, F3, 4, 28) (dual of [(398596, 4), 1594138, 29]-NRT-code), using
- OOA 4-folding [i] based on linear OA(3246, 1594384, F3, 28) (dual of [1594384, 1594138, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(3246, 1594386, F3, 28) (dual of [1594386, 1594140, 29]-code), using
- construction X applied to Ce(27) ⊂ Ce(21) [i] based on
- linear OA(3235, 1594323, F3, 28) (dual of [1594323, 1594088, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(3183, 1594323, F3, 22) (dual of [1594323, 1594140, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(311, 63, F3, 5) (dual of [63, 52, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(311, 85, F3, 5) (dual of [85, 74, 6]-code), using
- construction X applied to Ce(27) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(3246, 1594386, F3, 28) (dual of [1594386, 1594140, 29]-code), using
- OOA 4-folding [i] based on linear OA(3246, 1594384, F3, 28) (dual of [1594384, 1594138, 29]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3246, 398596, F3, 4, 28) (dual of [(398596, 4), 1594138, 29]-NRT-code), using
(219, 219+28, large)-Net in Base 3 — Upper bound on s
There is no (219, 247, large)-net in base 3, because
- 26 times m-reduction [i] would yield (219, 221, large)-net in base 3, but