Best Known (52−18, 52, s)-Nets in Base 49
(52−18, 52, 13072)-Net over F49 — Constructive and digital
Digital (34, 52, 13072)-net over F49, using
- net defined by OOA [i] based on linear OOA(4952, 13072, F49, 18, 18) (dual of [(13072, 18), 235244, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(4952, 117648, F49, 18) (dual of [117648, 117596, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(4952, 117649, F49, 18) (dual of [117649, 117597, 19]-code), using
- an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 117648 = 493−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- discarding factors / shortening the dual code based on linear OA(4952, 117649, F49, 18) (dual of [117649, 117597, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(4952, 117648, F49, 18) (dual of [117648, 117596, 19]-code), using
(52−18, 52, 57605)-Net over F49 — Digital
Digital (34, 52, 57605)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4952, 57605, F49, 2, 18) (dual of [(57605, 2), 115158, 19]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4952, 58826, F49, 2, 18) (dual of [(58826, 2), 117600, 19]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4952, 117652, F49, 18) (dual of [117652, 117600, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(16) [i] based on
- linear OA(4952, 117649, F49, 18) (dual of [117649, 117597, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 117648 = 493−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(4949, 117649, F49, 17) (dual of [117649, 117600, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 117648 = 493−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(490, 3, F49, 0) (dual of [3, 3, 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(17) ⊂ Ce(16) [i] based on
- OOA 2-folding [i] based on linear OA(4952, 117652, F49, 18) (dual of [117652, 117600, 19]-code), using
- discarding factors / shortening the dual code based on linear OOA(4952, 58826, F49, 2, 18) (dual of [(58826, 2), 117600, 19]-NRT-code), using
(52−18, 52, large)-Net in Base 49 — Upper bound on s
There is no (34, 52, large)-net in base 49, because
- 16 times m-reduction [i] would yield (34, 36, large)-net in base 49, but