Best Known (48−9, 48, s)-Nets in Base 49
(48−9, 48, 2882406)-Net over F49 — Constructive and digital
Digital (39, 48, 2882406)-net over F49, using
- (u, u+v)-construction [i] based on
- digital (9, 13, 2882402)-net over F49, using
- net defined by OOA [i] based on linear OOA(4913, 2882402, F49, 4, 4) (dual of [(2882402, 4), 11529595, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(4913, 2882402, F49, 3, 4) (dual of [(2882402, 3), 8647193, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(4913, 5764804, F49, 4) (dual of [5764804, 5764791, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(4913, 5764805, F49, 4) (dual of [5764805, 5764792, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- linear OA(4913, 5764801, F49, 4) (dual of [5764801, 5764788, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 494−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(499, 5764801, F49, 3) (dual of [5764801, 5764792, 4]-code or 5764801-cap in PG(8,49)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 494−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(490, 4, F49, 0) (dual of [4, 4, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(490, s, F49, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- discarding factors / shortening the dual code based on linear OA(4913, 5764805, F49, 4) (dual of [5764805, 5764792, 5]-code), using
- OA 2-folding and stacking [i] based on linear OA(4913, 5764804, F49, 4) (dual of [5764804, 5764791, 5]-code), using
- appending kth column [i] based on linear OOA(4913, 2882402, F49, 3, 4) (dual of [(2882402, 3), 8647193, 5]-NRT-code), using
- net defined by OOA [i] based on linear OOA(4913, 2882402, F49, 4, 4) (dual of [(2882402, 4), 11529595, 5]-NRT-code), using
- digital (26, 35, 1441203)-net over F49, using
- net defined by OOA [i] based on linear OOA(4935, 1441203, F49, 9, 9) (dual of [(1441203, 9), 12970792, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(4935, 5764813, F49, 9) (dual of [5764813, 5764778, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(4935, 5764815, F49, 9) (dual of [5764815, 5764780, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(5) [i] based on
- linear OA(4933, 5764801, F49, 9) (dual of [5764801, 5764768, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 494−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(4921, 5764801, F49, 6) (dual of [5764801, 5764780, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 494−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(492, 14, F49, 2) (dual of [14, 12, 3]-code or 14-arc in PG(1,49)), using
- discarding factors / shortening the dual code based on linear OA(492, 49, F49, 2) (dual of [49, 47, 3]-code or 49-arc in PG(1,49)), using
- Reed–Solomon code RS(47,49) [i]
- discarding factors / shortening the dual code based on linear OA(492, 49, F49, 2) (dual of [49, 47, 3]-code or 49-arc in PG(1,49)), using
- construction X applied to Ce(8) ⊂ Ce(5) [i] based on
- discarding factors / shortening the dual code based on linear OA(4935, 5764815, F49, 9) (dual of [5764815, 5764780, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(4935, 5764813, F49, 9) (dual of [5764813, 5764778, 10]-code), using
- net defined by OOA [i] based on linear OOA(4935, 1441203, F49, 9, 9) (dual of [(1441203, 9), 12970792, 10]-NRT-code), using
- digital (9, 13, 2882402)-net over F49, using
(48−9, 48, large)-Net over F49 — Digital
Digital (39, 48, large)-net over F49, using
- t-expansion [i] based on digital (37, 48, large)-net over F49, using
(48−9, 48, large)-Net in Base 49 — Upper bound on s
There is no (39, 48, large)-net in base 49, because
- 7 times m-reduction [i] would yield (39, 41, large)-net in base 49, but