Best Known (18, 18+10, s)-Nets in Base 49
(18, 18+10, 23530)-Net over F49 — Constructive and digital
Digital (18, 28, 23530)-net over F49, using
- net defined by OOA [i] based on linear OOA(4928, 23530, F49, 10, 10) (dual of [(23530, 10), 235272, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(4928, 117650, F49, 10) (dual of [117650, 117622, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(4928, 117652, F49, 10) (dual of [117652, 117624, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(8) [i] based on
- linear OA(4928, 117649, F49, 10) (dual of [117649, 117621, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 117648 = 493−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(4925, 117649, F49, 9) (dual of [117649, 117624, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 117648 = 493−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [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(9) ⊂ Ce(8) [i] based on
- discarding factors / shortening the dual code based on linear OA(4928, 117652, F49, 10) (dual of [117652, 117624, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(4928, 117650, F49, 10) (dual of [117650, 117622, 11]-code), using
(18, 18+10, 58826)-Net over F49 — Digital
Digital (18, 28, 58826)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4928, 58826, F49, 2, 10) (dual of [(58826, 2), 117624, 11]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4928, 117652, F49, 10) (dual of [117652, 117624, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(8) [i] based on
- linear OA(4928, 117649, F49, 10) (dual of [117649, 117621, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 117648 = 493−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(4925, 117649, F49, 9) (dual of [117649, 117624, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 117648 = 493−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [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(9) ⊂ Ce(8) [i] based on
- OOA 2-folding [i] based on linear OA(4928, 117652, F49, 10) (dual of [117652, 117624, 11]-code), using
(18, 18+10, large)-Net in Base 49 — Upper bound on s
There is no (18, 28, large)-net in base 49, because
- 8 times m-reduction [i] would yield (18, 20, large)-net in base 49, but