Best Known (41, 54, s)-Nets in Base 49
(41, 54, 960805)-Net over F49 — Constructive and digital
Digital (41, 54, 960805)-net over F49, using
- net defined by OOA [i] based on linear OOA(4954, 960805, F49, 13, 13) (dual of [(960805, 13), 12490411, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(4954, 5764831, F49, 13) (dual of [5764831, 5764777, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([0,3]) [i] based on
- linear OA(4949, 5764802, F49, 13) (dual of [5764802, 5764753, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 5764802 | 498−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(4925, 5764802, F49, 7) (dual of [5764802, 5764777, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 5764802 | 498−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- linear OA(495, 29, F49, 5) (dual of [29, 24, 6]-code or 29-arc in PG(4,49)), using
- discarding factors / shortening the dual code based on linear OA(495, 49, F49, 5) (dual of [49, 44, 6]-code or 49-arc in PG(4,49)), using
- Reed–Solomon code RS(44,49) [i]
- discarding factors / shortening the dual code based on linear OA(495, 49, F49, 5) (dual of [49, 44, 6]-code or 49-arc in PG(4,49)), using
- construction X applied to C([0,6]) ⊂ C([0,3]) [i] based on
- OOA 6-folding and stacking with additional row [i] based on linear OA(4954, 5764831, F49, 13) (dual of [5764831, 5764777, 14]-code), using
(41, 54, 5764831)-Net over F49 — Digital
Digital (41, 54, 5764831)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4954, 5764831, F49, 13) (dual of [5764831, 5764777, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([0,3]) [i] based on
- linear OA(4949, 5764802, F49, 13) (dual of [5764802, 5764753, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 5764802 | 498−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(4925, 5764802, F49, 7) (dual of [5764802, 5764777, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 5764802 | 498−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- linear OA(495, 29, F49, 5) (dual of [29, 24, 6]-code or 29-arc in PG(4,49)), using
- discarding factors / shortening the dual code based on linear OA(495, 49, F49, 5) (dual of [49, 44, 6]-code or 49-arc in PG(4,49)), using
- Reed–Solomon code RS(44,49) [i]
- discarding factors / shortening the dual code based on linear OA(495, 49, F49, 5) (dual of [49, 44, 6]-code or 49-arc in PG(4,49)), using
- construction X applied to C([0,6]) ⊂ C([0,3]) [i] based on
(41, 54, large)-Net in Base 49 — Upper bound on s
There is no (41, 54, large)-net in base 49, because
- 11 times m-reduction [i] would yield (41, 43, large)-net in base 49, but