Best Known (214−25, 214, s)-Nets in Base 3
(214−25, 214, 132862)-Net over F3 — Constructive and digital
Digital (189, 214, 132862)-net over F3, using
- 31 times duplication [i] based on digital (188, 213, 132862)-net over F3, using
- net defined by OOA [i] based on linear OOA(3213, 132862, F3, 25, 25) (dual of [(132862, 25), 3321337, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(3213, 1594345, F3, 25) (dual of [1594345, 1594132, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(3213, 1594353, F3, 25) (dual of [1594353, 1594140, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(21) [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(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(34, 30, F3, 2) (dual of [30, 26, 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(24) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(3213, 1594353, F3, 25) (dual of [1594353, 1594140, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(3213, 1594345, F3, 25) (dual of [1594345, 1594132, 26]-code), using
- net defined by OOA [i] based on linear OOA(3213, 132862, F3, 25, 25) (dual of [(132862, 25), 3321337, 26]-NRT-code), using
(214−25, 214, 398588)-Net over F3 — Digital
Digital (189, 214, 398588)-net over F3, using
- 31 times duplication [i] based on digital (188, 213, 398588)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3213, 398588, F3, 4, 25) (dual of [(398588, 4), 1594139, 26]-NRT-code), using
- OOA 4-folding [i] based on linear OA(3213, 1594352, F3, 25) (dual of [1594352, 1594139, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(3213, 1594353, F3, 25) (dual of [1594353, 1594140, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(21) [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(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(34, 30, F3, 2) (dual of [30, 26, 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(24) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(3213, 1594353, F3, 25) (dual of [1594353, 1594140, 26]-code), using
- OOA 4-folding [i] based on linear OA(3213, 1594352, F3, 25) (dual of [1594352, 1594139, 26]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3213, 398588, F3, 4, 25) (dual of [(398588, 4), 1594139, 26]-NRT-code), using
(214−25, 214, large)-Net in Base 3 — Upper bound on s
There is no (189, 214, large)-net in base 3, because
- 23 times m-reduction [i] would yield (189, 191, large)-net in base 3, but