Best Known (53−14, 53, s)-Nets in Base 49
(53−14, 53, 823543)-Net over F49 — Constructive and digital
Digital (39, 53, 823543)-net over F49, using
- net defined by OOA [i] based on linear OOA(4953, 823543, F49, 14, 14) (dual of [(823543, 14), 11529549, 15]-NRT-code), using
- OA 7-folding and stacking [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]
- OA 7-folding and stacking [i] based on linear OA(4953, 5764801, F49, 14) (dual of [5764801, 5764748, 15]-code), using
(53−14, 53, 2882402)-Net over F49 — Digital
Digital (39, 53, 2882402)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4953, 2882402, F49, 2, 14) (dual of [(2882402, 2), 5764751, 15]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4953, 5764804, F49, 14) (dual of [5764804, 5764751, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(4953, 5764805, F49, 14) (dual of [5764805, 5764752, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(12) [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(4949, 5764801, F49, 13) (dual of [5764801, 5764752, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 494−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [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(13) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(4953, 5764805, F49, 14) (dual of [5764805, 5764752, 15]-code), using
- OOA 2-folding [i] based on linear OA(4953, 5764804, F49, 14) (dual of [5764804, 5764751, 15]-code), using
(53−14, 53, large)-Net in Base 49 — Upper bound on s
There is no (39, 53, large)-net in base 49, because
- 12 times m-reduction [i] would yield (39, 41, large)-net in base 49, but