Best Known (41, 55, s)-Nets in Base 49
(41, 55, 823545)-Net over F49 — Constructive and digital
Digital (41, 55, 823545)-net over F49, using
- net defined by OOA [i] based on linear OOA(4955, 823545, F49, 14, 14) (dual of [(823545, 14), 11529575, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(4955, 5764815, F49, 14) (dual of [5764815, 5764760, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(10) [i] based on
- linear OA(4953, 5764801, F49, 14) (dual of [5764801, 5764748, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 494−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(4941, 5764801, F49, 11) (dual of [5764801, 5764760, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 494−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [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(13) ⊂ Ce(10) [i] based on
- OA 7-folding and stacking [i] based on linear OA(4955, 5764815, F49, 14) (dual of [5764815, 5764760, 15]-code), using
(41, 55, 4446333)-Net over F49 — Digital
Digital (41, 55, 4446333)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4955, 4446333, F49, 14) (dual of [4446333, 4446278, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(4955, 5764815, F49, 14) (dual of [5764815, 5764760, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(10) [i] based on
- linear OA(4953, 5764801, F49, 14) (dual of [5764801, 5764748, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 494−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(4941, 5764801, F49, 11) (dual of [5764801, 5764760, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 494−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [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(13) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(4955, 5764815, F49, 14) (dual of [5764815, 5764760, 15]-code), using
(41, 55, large)-Net in Base 49 — Upper bound on s
There is no (41, 55, large)-net in base 49, because
- 12 times m-reduction [i] would yield (41, 43, large)-net in base 49, but