Best Known (50−12, 50, s)-Nets in Base 49
(50−12, 50, 960805)-Net over F49 — Constructive and digital
Digital (38, 50, 960805)-net over F49, using
- net defined by OOA [i] based on linear OOA(4950, 960805, F49, 12, 12) (dual of [(960805, 12), 11529610, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(4950, 5764830, F49, 12) (dual of [5764830, 5764780, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(5) [i] based on
- linear OA(4945, 5764801, F49, 12) (dual of [5764801, 5764756, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 494−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [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(495, 29, F49, 5) (dual of [29, 24, 6]-code or 29-arc in PG(4,49)), using
- discarding factors / shortening the dual code based on linear OA(495, 49, F49, 5) (dual of [49, 44, 6]-code or 49-arc in PG(4,49)), using
- Reed–Solomon code RS(44,49) [i]
- discarding factors / shortening the dual code based on linear OA(495, 49, F49, 5) (dual of [49, 44, 6]-code or 49-arc in PG(4,49)), using
- construction X applied to Ce(11) ⊂ Ce(5) [i] based on
- OA 6-folding and stacking [i] based on linear OA(4950, 5764830, F49, 12) (dual of [5764830, 5764780, 13]-code), using
(50−12, 50, 5764830)-Net over F49 — Digital
Digital (38, 50, 5764830)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4950, 5764830, F49, 12) (dual of [5764830, 5764780, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(5) [i] based on
- linear OA(4945, 5764801, F49, 12) (dual of [5764801, 5764756, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 494−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [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(495, 29, F49, 5) (dual of [29, 24, 6]-code or 29-arc in PG(4,49)), using
- discarding factors / shortening the dual code based on linear OA(495, 49, F49, 5) (dual of [49, 44, 6]-code or 49-arc in PG(4,49)), using
- Reed–Solomon code RS(44,49) [i]
- discarding factors / shortening the dual code based on linear OA(495, 49, F49, 5) (dual of [49, 44, 6]-code or 49-arc in PG(4,49)), using
- construction X applied to Ce(11) ⊂ Ce(5) [i] based on
(50−12, 50, large)-Net in Base 49 — Upper bound on s
There is no (38, 50, large)-net in base 49, because
- 10 times m-reduction [i] would yield (38, 40, large)-net in base 49, but