Best Known (55−9, 55, s)-Nets in Base 49
(55−9, 55, 4323653)-Net over F49 — Constructive and digital
Digital (46, 55, 4323653)-net over F49, using
- generalized (u, u+v)-construction [i] based on
- digital (0, 2, 50)-net over F49, using
- digital (4, 7, 1441201)-net over F49, using
- s-reduction based on digital (4, 7, 5769603)-net over F49, using
- net defined by OOA [i] based on linear OOA(497, 5769603, F49, 3, 3) (dual of [(5769603, 3), 17308802, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(497, 5769603, F49, 2, 3) (dual of [(5769603, 2), 11539199, 4]-NRT-code), using
- net defined by OOA [i] based on linear OOA(497, 5769603, F49, 3, 3) (dual of [(5769603, 3), 17308802, 4]-NRT-code), using
- s-reduction based on digital (4, 7, 5769603)-net over F49, using
- digital (9, 13, 1441201)-net over F49, using
- s-reduction 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
- s-reduction based on digital (9, 13, 2882402)-net over F49, using
- digital (24, 33, 1441201)-net over F49, using
- net defined by OOA [i] based on linear OOA(4933, 1441201, F49, 9, 9) (dual of [(1441201, 9), 12970776, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(4933, 5764805, F49, 9) (dual of [5764805, 5764772, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(7) [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(4929, 5764801, F49, 8) (dual of [5764801, 5764772, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 494−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(490, 4, F49, 0) (dual of [4, 4, 1]-code) (see above)
- construction X applied to Ce(8) ⊂ Ce(7) [i] based on
- OOA 4-folding and stacking with additional row [i] based on linear OA(4933, 5764805, F49, 9) (dual of [5764805, 5764772, 10]-code), using
- net defined by OOA [i] based on linear OOA(4933, 1441201, F49, 9, 9) (dual of [(1441201, 9), 12970776, 10]-NRT-code), using
(55−9, 55, large)-Net over F49 — Digital
Digital (46, 55, large)-net over F49, using
- 493 times duplication [i] based on digital (43, 52, large)-net over F49, using
- t-expansion [i] based on digital (40, 52, large)-net over F49, using
(55−9, 55, large)-Net in Base 49 — Upper bound on s
There is no (46, 55, large)-net in base 49, because
- 7 times m-reduction [i] would yield (46, 48, large)-net in base 49, but