Best Known (48, 48+8, s)-Nets in Base 3
(48, 48+8, 44289)-Net over F3 — Constructive and digital
Digital (48, 56, 44289)-net over F3, using
- net defined by OOA [i] based on linear OOA(356, 44289, F3, 8, 8) (dual of [(44289, 8), 354256, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(356, 177156, F3, 8) (dual of [177156, 177100, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(356, 177158, F3, 8) (dual of [177158, 177102, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(6) [i] based on
- linear OA(356, 177147, F3, 8) (dual of [177147, 177091, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(345, 177147, F3, 7) (dual of [177147, 177102, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(30, 11, F3, 0) (dual of [11, 11, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(7) ⊂ Ce(6) [i] based on
- discarding factors / shortening the dual code based on linear OA(356, 177158, F3, 8) (dual of [177158, 177102, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(356, 177156, F3, 8) (dual of [177156, 177100, 9]-code), using
(48, 48+8, 88579)-Net over F3 — Digital
Digital (48, 56, 88579)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(356, 88579, F3, 2, 8) (dual of [(88579, 2), 177102, 9]-NRT-code), using
- OOA 2-folding [i] based on linear OA(356, 177158, F3, 8) (dual of [177158, 177102, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(6) [i] based on
- linear OA(356, 177147, F3, 8) (dual of [177147, 177091, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(345, 177147, F3, 7) (dual of [177147, 177102, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(30, 11, F3, 0) (dual of [11, 11, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(7) ⊂ Ce(6) [i] based on
- OOA 2-folding [i] based on linear OA(356, 177158, F3, 8) (dual of [177158, 177102, 9]-code), using
(48, 48+8, 5293221)-Net in Base 3 — Upper bound on s
There is no (48, 56, 5293222)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 523 347706 275870 513453 988617 > 356 [i]