Best Known (44−11, 44, s)-Nets in Base 49
(44−11, 44, 1152964)-Net over F49 — Constructive and digital
Digital (33, 44, 1152964)-net over F49, using
- net defined by OOA [i] based on linear OOA(4944, 1152964, F49, 11, 11) (dual of [(1152964, 11), 12682560, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(4944, 5764821, F49, 11) (dual of [5764821, 5764777, 12]-code), using
- construction X applied to C([0,5]) ⊂ C([0,3]) [i] based on
- linear OA(4941, 5764802, F49, 11) (dual of [5764802, 5764761, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 5764802 | 498−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (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(493, 19, F49, 3) (dual of [19, 16, 4]-code or 19-arc in PG(2,49) or 19-cap in PG(2,49)), using
- discarding factors / shortening the dual code based on linear OA(493, 49, F49, 3) (dual of [49, 46, 4]-code or 49-arc in PG(2,49) or 49-cap in PG(2,49)), using
- Reed–Solomon code RS(46,49) [i]
- discarding factors / shortening the dual code based on linear OA(493, 49, F49, 3) (dual of [49, 46, 4]-code or 49-arc in PG(2,49) or 49-cap in PG(2,49)), using
- construction X applied to C([0,5]) ⊂ C([0,3]) [i] based on
- OOA 5-folding and stacking with additional row [i] based on linear OA(4944, 5764821, F49, 11) (dual of [5764821, 5764777, 12]-code), using
(44−11, 44, 5764821)-Net over F49 — Digital
Digital (33, 44, 5764821)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4944, 5764821, F49, 11) (dual of [5764821, 5764777, 12]-code), using
- construction X applied to C([0,5]) ⊂ C([0,3]) [i] based on
- linear OA(4941, 5764802, F49, 11) (dual of [5764802, 5764761, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 5764802 | 498−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (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(493, 19, F49, 3) (dual of [19, 16, 4]-code or 19-arc in PG(2,49) or 19-cap in PG(2,49)), using
- discarding factors / shortening the dual code based on linear OA(493, 49, F49, 3) (dual of [49, 46, 4]-code or 49-arc in PG(2,49) or 49-cap in PG(2,49)), using
- Reed–Solomon code RS(46,49) [i]
- discarding factors / shortening the dual code based on linear OA(493, 49, F49, 3) (dual of [49, 46, 4]-code or 49-arc in PG(2,49) or 49-cap in PG(2,49)), using
- construction X applied to C([0,5]) ⊂ C([0,3]) [i] based on
(44−11, 44, large)-Net in Base 49 — Upper bound on s
There is no (33, 44, large)-net in base 49, because
- 9 times m-reduction [i] would yield (33, 35, large)-net in base 49, but