Best Known (52−13, 52, s)-Nets in Base 49
(52−13, 52, 960803)-Net over F49 — Constructive and digital
Digital (39, 52, 960803)-net over F49, using
- net defined by OOA [i] based on linear OOA(4952, 960803, F49, 13, 13) (dual of [(960803, 13), 12490387, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(4952, 5764819, F49, 13) (dual of [5764819, 5764767, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(4952, 5764821, F49, 13) (dual of [5764821, 5764769, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([0,4]) [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(4933, 5764802, F49, 9) (dual of [5764802, 5764769, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 5764802 | 498−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (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([0,6]) ⊂ C([0,4]) [i] based on
- discarding factors / shortening the dual code based on linear OA(4952, 5764821, F49, 13) (dual of [5764821, 5764769, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(4952, 5764819, F49, 13) (dual of [5764819, 5764767, 14]-code), using
(52−13, 52, 5764821)-Net over F49 — Digital
Digital (39, 52, 5764821)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4952, 5764821, F49, 13) (dual of [5764821, 5764769, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([0,4]) [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(4933, 5764802, F49, 9) (dual of [5764802, 5764769, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 5764802 | 498−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (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([0,6]) ⊂ C([0,4]) [i] based on
(52−13, 52, large)-Net in Base 49 — Upper bound on s
There is no (39, 52, large)-net in base 49, because
- 11 times m-reduction [i] would yield (39, 41, large)-net in base 49, but