Best Known (47, 47+7, s)-Nets in Base 3
(47, 47+7, 531445)-Net over F3 — Constructive and digital
Digital (47, 54, 531445)-net over F3, using
- net defined by OOA [i] based on linear OOA(354, 531445, F3, 7, 7) (dual of [(531445, 7), 3720061, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(354, 1594336, F3, 7) (dual of [1594336, 1594282, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(354, 1594337, F3, 7) (dual of [1594337, 1594283, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(4) [i] based on
- linear OA(353, 1594323, F3, 7) (dual of [1594323, 1594270, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(340, 1594323, F3, 5) (dual of [1594323, 1594283, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(31, 14, F3, 1) (dual of [14, 13, 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 Ce(6) ⊂ Ce(4) [i] based on
- discarding factors / shortening the dual code based on linear OA(354, 1594337, F3, 7) (dual of [1594337, 1594283, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(354, 1594336, F3, 7) (dual of [1594336, 1594282, 8]-code), using
(47, 47+7, 797169)-Net over F3 — Digital
Digital (47, 54, 797169)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(354, 797169, F3, 2, 7) (dual of [(797169, 2), 1594284, 8]-NRT-code), using
- OOA 2-folding [i] based on linear OA(354, 1594338, F3, 7) (dual of [1594338, 1594284, 8]-code), using
- construction X4 applied to Ce(6) ⊂ Ce(4) [i] based on
- linear OA(353, 1594323, F3, 7) (dual of [1594323, 1594270, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(340, 1594323, F3, 5) (dual of [1594323, 1594283, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(314, 15, F3, 14) (dual of [15, 1, 15]-code or 15-arc in PG(13,3)), using
- dual of repetition code with length 15 [i]
- linear OA(31, 15, F3, 1) (dual of [15, 14, 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 X4 applied to Ce(6) ⊂ Ce(4) [i] based on
- OOA 2-folding [i] based on linear OA(354, 1594338, F3, 7) (dual of [1594338, 1594284, 8]-code), using
(47, 47+7, large)-Net in Base 3 — Upper bound on s
There is no (47, 54, large)-net in base 3, because
- 5 times m-reduction [i] would yield (47, 49, large)-net in base 3, but