Best Known (38, 38+9, s)-Nets in Base 49
(38, 38+9, 2882404)-Net over F49 — Constructive and digital
Digital (38, 47, 2882404)-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 (25, 34, 1441202)-net over F49, using
- net defined by OOA [i] based on linear OOA(4934, 1441202, F49, 9, 9) (dual of [(1441202, 9), 12970784, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(4934, 5764809, F49, 9) (dual of [5764809, 5764775, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(4934, 5764811, F49, 9) (dual of [5764811, 5764777, 10]-code), using
- construction X applied to C([0,4]) ⊂ C([0,3]) [i] based on
- 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(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(491, 9, F49, 1) (dual of [9, 8, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(491, s, F49, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,4]) ⊂ C([0,3]) [i] based on
- discarding factors / shortening the dual code based on linear OA(4934, 5764811, F49, 9) (dual of [5764811, 5764777, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(4934, 5764809, F49, 9) (dual of [5764809, 5764775, 10]-code), using
- net defined by OOA [i] based on linear OOA(4934, 1441202, F49, 9, 9) (dual of [(1441202, 9), 12970784, 10]-NRT-code), using
- digital (9, 13, 2882402)-net over F49, using
(38, 38+9, large)-Net over F49 — Digital
Digital (38, 47, large)-net over F49, using
- t-expansion [i] based on digital (37, 47, large)-net over F49, using
- 1 times m-reduction [i] based on digital (37, 48, large)-net over F49, using
(38, 38+9, large)-Net in Base 49 — Upper bound on s
There is no (38, 47, large)-net in base 49, because
- 7 times m-reduction [i] would yield (38, 40, large)-net in base 49, but