Best Known (41−10, 41, s)-Nets in Base 49
(41−10, 41, 1152965)-Net over F49 — Constructive and digital
Digital (31, 41, 1152965)-net over F49, using
- net defined by OOA [i] based on linear OOA(4941, 1152965, F49, 10, 10) (dual of [(1152965, 10), 11529609, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(4941, 5764825, F49, 10) (dual of [5764825, 5764784, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(4) [i] based on
- linear OA(4937, 5764801, F49, 10) (dual of [5764801, 5764764, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 494−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(4917, 5764801, F49, 5) (dual of [5764801, 5764784, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 494−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(494, 24, F49, 4) (dual of [24, 20, 5]-code or 24-arc in PG(3,49)), using
- discarding factors / shortening the dual code based on linear OA(494, 49, F49, 4) (dual of [49, 45, 5]-code or 49-arc in PG(3,49)), using
- Reed–Solomon code RS(45,49) [i]
- discarding factors / shortening the dual code based on linear OA(494, 49, F49, 4) (dual of [49, 45, 5]-code or 49-arc in PG(3,49)), using
- construction X applied to Ce(9) ⊂ Ce(4) [i] based on
- OA 5-folding and stacking [i] based on linear OA(4941, 5764825, F49, 10) (dual of [5764825, 5764784, 11]-code), using
(41−10, 41, 5764825)-Net over F49 — Digital
Digital (31, 41, 5764825)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4941, 5764825, F49, 10) (dual of [5764825, 5764784, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(4) [i] based on
- linear OA(4937, 5764801, F49, 10) (dual of [5764801, 5764764, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 494−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(4917, 5764801, F49, 5) (dual of [5764801, 5764784, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 494−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(494, 24, F49, 4) (dual of [24, 20, 5]-code or 24-arc in PG(3,49)), using
- discarding factors / shortening the dual code based on linear OA(494, 49, F49, 4) (dual of [49, 45, 5]-code or 49-arc in PG(3,49)), using
- Reed–Solomon code RS(45,49) [i]
- discarding factors / shortening the dual code based on linear OA(494, 49, F49, 4) (dual of [49, 45, 5]-code or 49-arc in PG(3,49)), using
- construction X applied to Ce(9) ⊂ Ce(4) [i] based on
(41−10, 41, large)-Net in Base 49 — Upper bound on s
There is no (31, 41, large)-net in base 49, because
- 8 times m-reduction [i] would yield (31, 33, large)-net in base 49, but