Best Known (30, 30+14, s)-Nets in Base 49
(30, 30+14, 16809)-Net over F49 — Constructive and digital
Digital (30, 44, 16809)-net over F49, using
- 491 times duplication [i] based on digital (29, 43, 16809)-net over F49, using
- net defined by OOA [i] based on linear OOA(4943, 16809, F49, 14, 14) (dual of [(16809, 14), 235283, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(4943, 117663, F49, 14) (dual of [117663, 117620, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(4943, 117664, F49, 14) (dual of [117664, 117621, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(9) [i] based on
- linear OA(4940, 117649, F49, 14) (dual of [117649, 117609, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 117648 = 493−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- 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(493, 15, F49, 3) (dual of [15, 12, 4]-code or 15-arc in PG(2,49) or 15-cap in PG(2,49)), using
- discarding factors / shortening the dual code based on linear OA(493, 49, F49, 3) (dual of [49, 46, 4]-code or 49-arc in PG(2,49) or 49-cap in PG(2,49)), using
- Reed–Solomon code RS(46,49) [i]
- discarding factors / shortening the dual code based on linear OA(493, 49, F49, 3) (dual of [49, 46, 4]-code or 49-arc in PG(2,49) or 49-cap in PG(2,49)), using
- construction X applied to Ce(13) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(4943, 117664, F49, 14) (dual of [117664, 117621, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(4943, 117663, F49, 14) (dual of [117663, 117620, 15]-code), using
- net defined by OOA [i] based on linear OOA(4943, 16809, F49, 14, 14) (dual of [(16809, 14), 235283, 15]-NRT-code), using
(30, 30+14, 117668)-Net over F49 — Digital
Digital (30, 44, 117668)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4944, 117668, F49, 14) (dual of [117668, 117624, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(8) [i] based on
- linear OA(4940, 117649, F49, 14) (dual of [117649, 117609, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 117648 = 493−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [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(494, 19, F49, 4) (dual of [19, 15, 5]-code or 19-arc in PG(3,49)), using
- discarding factors / shortening the dual code based on linear OA(494, 49, F49, 4) (dual of [49, 45, 5]-code or 49-arc in PG(3,49)), using
- Reed–Solomon code RS(45,49) [i]
- discarding factors / shortening the dual code based on linear OA(494, 49, F49, 4) (dual of [49, 45, 5]-code or 49-arc in PG(3,49)), using
- construction X applied to Ce(13) ⊂ Ce(8) [i] based on
(30, 30+14, large)-Net in Base 49 — Upper bound on s
There is no (30, 44, large)-net in base 49, because
- 12 times m-reduction [i] would yield (30, 32, large)-net in base 49, but