Best Known (36, 36+19, s)-Nets in Base 49
(36, 36+19, 13072)-Net over F49 — Constructive and digital
Digital (36, 55, 13072)-net over F49, using
- net defined by OOA [i] based on linear OOA(4955, 13072, F49, 19, 19) (dual of [(13072, 19), 248313, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(4955, 117649, F49, 19) (dual of [117649, 117594, 20]-code), using
- an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 117648 = 493−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- OOA 9-folding and stacking with additional row [i] based on linear OA(4955, 117649, F49, 19) (dual of [117649, 117594, 20]-code), using
(36, 36+19, 56236)-Net over F49 — Digital
Digital (36, 55, 56236)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4955, 56236, F49, 2, 19) (dual of [(56236, 2), 112417, 20]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4955, 58826, F49, 2, 19) (dual of [(58826, 2), 117597, 20]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4955, 117652, F49, 19) (dual of [117652, 117597, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(17) [i] based on
- linear OA(4955, 117649, F49, 19) (dual of [117649, 117594, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 117648 = 493−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- 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(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(18) ⊂ Ce(17) [i] based on
- OOA 2-folding [i] based on linear OA(4955, 117652, F49, 19) (dual of [117652, 117597, 20]-code), using
- discarding factors / shortening the dual code based on linear OOA(4955, 58826, F49, 2, 19) (dual of [(58826, 2), 117597, 20]-NRT-code), using
(36, 36+19, large)-Net in Base 49 — Upper bound on s
There is no (36, 55, large)-net in base 49, because
- 17 times m-reduction [i] would yield (36, 38, large)-net in base 49, but