Best Known (42−13, 42, s)-Nets in Base 49
(42−13, 42, 19612)-Net over F49 — Constructive and digital
Digital (29, 42, 19612)-net over F49, using
- net defined by OOA [i] based on linear OOA(4942, 19612, F49, 13, 13) (dual of [(19612, 13), 254914, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(4942, 117673, F49, 13) (dual of [117673, 117631, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([0,3]) [i] based on
- linear OA(4937, 117650, F49, 13) (dual of [117650, 117613, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 117650 | 496−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(4919, 117650, F49, 7) (dual of [117650, 117631, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 117650 | 496−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- linear OA(495, 23, F49, 5) (dual of [23, 18, 6]-code or 23-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(4942, 117673, F49, 13) (dual of [117673, 117631, 14]-code), using
(42−13, 42, 117673)-Net over F49 — Digital
Digital (29, 42, 117673)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4942, 117673, F49, 13) (dual of [117673, 117631, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([0,3]) [i] based on
- linear OA(4937, 117650, F49, 13) (dual of [117650, 117613, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 117650 | 496−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(4919, 117650, F49, 7) (dual of [117650, 117631, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 117650 | 496−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- linear OA(495, 23, F49, 5) (dual of [23, 18, 6]-code or 23-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
(42−13, 42, large)-Net in Base 49 — Upper bound on s
There is no (29, 42, large)-net in base 49, because
- 11 times m-reduction [i] would yield (29, 31, large)-net in base 49, but