Best Known (49, 49+7, s)-Nets in Base 3
(49, 49+7, 531446)-Net over F3 — Constructive and digital
Digital (49, 56, 531446)-net over F3, using
- 31 times duplication [i] based on digital (48, 55, 531446)-net over F3, using
- net defined by OOA [i] based on linear OOA(355, 531446, F3, 7, 7) (dual of [(531446, 7), 3720067, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(355, 1594339, F3, 7) (dual of [1594339, 1594284, 8]-code), using
- 1 times code embedding in larger space [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
- 1 times code embedding in larger space [i] based on linear OA(354, 1594338, F3, 7) (dual of [1594338, 1594284, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(355, 1594339, F3, 7) (dual of [1594339, 1594284, 8]-code), using
- net defined by OOA [i] based on linear OOA(355, 531446, F3, 7, 7) (dual of [(531446, 7), 3720067, 8]-NRT-code), using
(49, 49+7, 797170)-Net over F3 — Digital
Digital (49, 56, 797170)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(356, 797170, F3, 2, 7) (dual of [(797170, 2), 1594284, 8]-NRT-code), using
- 1 times NRT-code embedding in larger space [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
- 1 times NRT-code embedding in larger space [i] based on linear OOA(354, 797169, F3, 2, 7) (dual of [(797169, 2), 1594284, 8]-NRT-code), using
(49, 49+7, large)-Net in Base 3 — Upper bound on s
There is no (49, 56, large)-net in base 3, because
- 5 times m-reduction [i] would yield (49, 51, large)-net in base 3, but