Best Known (48, 54, s)-Nets in Base 3
(48, 54, 531450)-Net over F3 — Constructive and digital
Digital (48, 54, 531450)-net over F3, using
- net defined by OOA [i] based on linear OOA(354, 531450, F3, 6, 6) (dual of [(531450, 6), 3188646, 7]-NRT-code), using
- appending kth column [i] based on linear OOA(354, 531450, F3, 5, 6) (dual of [(531450, 5), 2657196, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(354, 1594350, F3, 6) (dual of [1594350, 1594296, 7]-code), using
- construction X applied to Ce(6) ⊂ Ce(3) [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(327, 1594323, F3, 4) (dual of [1594323, 1594296, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(31, 27, F3, 1) (dual of [27, 26, 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(3) [i] based on
- OA 3-folding and stacking [i] based on linear OA(354, 1594350, F3, 6) (dual of [1594350, 1594296, 7]-code), using
- appending kth column [i] based on linear OOA(354, 531450, F3, 5, 6) (dual of [(531450, 5), 2657196, 7]-NRT-code), using
(48, 54, 1594351)-Net over F3 — Digital
Digital (48, 54, 1594351)-net over F3, using
- net defined by OOA [i] based on linear OOA(354, 1594351, F3, 6, 6) (dual of [(1594351, 6), 9566052, 7]-NRT-code), using
- appending kth column [i] based on linear OOA(354, 1594351, F3, 5, 6) (dual of [(1594351, 5), 7971701, 7]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(354, 1594351, F3, 6) (dual of [1594351, 1594297, 7]-code), using
- construction X4 applied to Ce(6) ⊂ Ce(3) [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(327, 1594323, F3, 4) (dual of [1594323, 1594296, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(327, 28, F3, 27) (dual of [28, 1, 28]-code or 28-arc in PG(26,3)), using
- dual of repetition code with length 28 [i]
- linear OA(31, 28, F3, 1) (dual of [28, 27, 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(3) [i] based on
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(354, 1594351, F3, 6) (dual of [1594351, 1594297, 7]-code), using
- appending kth column [i] based on linear OOA(354, 1594351, F3, 5, 6) (dual of [(1594351, 5), 7971701, 7]-NRT-code), using
(48, 54, large)-Net in Base 3 — Upper bound on s
There is no (48, 54, large)-net in base 3, because
- 4 times m-reduction [i] would yield (48, 50, large)-net in base 3, but