Best Known (21, 21+8, s)-Nets in Base 49
(21, 21+8, 1441201)-Net over F49 — Constructive and digital
Digital (21, 29, 1441201)-net over F49, using
- net defined by OOA [i] based on linear OOA(4929, 1441201, F49, 8, 8) (dual of [(1441201, 8), 11529579, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(4929, 5764804, F49, 8) (dual of [5764804, 5764775, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(4929, 5764805, F49, 8) (dual of [5764805, 5764776, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(6) [i] based on
- 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(4925, 5764801, F49, 7) (dual of [5764801, 5764776, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 494−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [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(7) ⊂ Ce(6) [i] based on
- discarding factors / shortening the dual code based on linear OA(4929, 5764805, F49, 8) (dual of [5764805, 5764776, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(4929, 5764804, F49, 8) (dual of [5764804, 5764775, 9]-code), using
(21, 21+8, 4814624)-Net over F49 — Digital
Digital (21, 29, 4814624)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4929, 4814624, F49, 8) (dual of [4814624, 4814595, 9]-code), using
- discarding factors / shortening the dual code based on 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]
- discarding factors / shortening the dual code based on linear OA(4929, 5764801, F49, 8) (dual of [5764801, 5764772, 9]-code), using
(21, 21+8, large)-Net in Base 49 — Upper bound on s
There is no (21, 29, large)-net in base 49, because
- 6 times m-reduction [i] would yield (21, 23, large)-net in base 49, but