Best Known (19−7, 19, s)-Nets in Base 49
(19−7, 19, 39217)-Net over F49 — Constructive and digital
Digital (12, 19, 39217)-net over F49, using
- net defined by OOA [i] based on linear OOA(4919, 39217, F49, 7, 7) (dual of [(39217, 7), 274500, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(4919, 117652, F49, 7) (dual of [117652, 117633, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- linear OA(4919, 117649, F49, 7) (dual of [117649, 117630, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 117648 = 493−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(4916, 117649, F49, 6) (dual of [117649, 117633, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 117648 = 493−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [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(6) ⊂ Ce(5) [i] based on
- OOA 3-folding and stacking with additional row [i] based on linear OA(4919, 117652, F49, 7) (dual of [117652, 117633, 8]-code), using
(19−7, 19, 65961)-Net over F49 — Digital
Digital (12, 19, 65961)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4919, 65961, F49, 7) (dual of [65961, 65942, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(4919, 117649, F49, 7) (dual of [117649, 117630, 8]-code), using
- an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 117648 = 493−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- discarding factors / shortening the dual code based on linear OA(4919, 117649, F49, 7) (dual of [117649, 117630, 8]-code), using
(19−7, 19, large)-Net in Base 49 — Upper bound on s
There is no (12, 19, large)-net in base 49, because
- 5 times m-reduction [i] would yield (12, 14, large)-net in base 49, but