Best Known (38, 38+17, s)-Nets in Base 49
(38, 38+17, 14709)-Net over F49 — Constructive and digital
Digital (38, 55, 14709)-net over F49, using
- 491 times duplication [i] based on digital (37, 54, 14709)-net over F49, using
- net defined by OOA [i] based on linear OOA(4954, 14709, F49, 17, 17) (dual of [(14709, 17), 249999, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(4954, 117673, F49, 17) (dual of [117673, 117619, 18]-code), using
- construction X applied to C([0,8]) ⊂ C([0,5]) [i] based on
- linear OA(4949, 117650, F49, 17) (dual of [117650, 117601, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 117650 | 496−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(4931, 117650, F49, 11) (dual of [117650, 117619, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 117650 | 496−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(495, 23, F49, 5) (dual of [23, 18, 6]-code or 23-arc in PG(4,49)), using
- discarding factors / shortening the dual code based on linear OA(495, 49, F49, 5) (dual of [49, 44, 6]-code or 49-arc in PG(4,49)), using
- Reed–Solomon code RS(44,49) [i]
- discarding factors / shortening the dual code based on linear OA(495, 49, F49, 5) (dual of [49, 44, 6]-code or 49-arc in PG(4,49)), using
- construction X applied to C([0,8]) ⊂ C([0,5]) [i] based on
- OOA 8-folding and stacking with additional row [i] based on linear OA(4954, 117673, F49, 17) (dual of [117673, 117619, 18]-code), using
- net defined by OOA [i] based on linear OOA(4954, 14709, F49, 17, 17) (dual of [(14709, 17), 249999, 18]-NRT-code), using
(38, 38+17, 117676)-Net over F49 — Digital
Digital (38, 55, 117676)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4955, 117676, F49, 17) (dual of [117676, 117621, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(9) [i] based on
- 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(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(496, 27, F49, 6) (dual of [27, 21, 7]-code or 27-arc in PG(5,49)), using
- discarding factors / shortening the dual code based on linear OA(496, 49, F49, 6) (dual of [49, 43, 7]-code or 49-arc in PG(5,49)), using
- Reed–Solomon code RS(43,49) [i]
- discarding factors / shortening the dual code based on linear OA(496, 49, F49, 6) (dual of [49, 43, 7]-code or 49-arc in PG(5,49)), using
- construction X applied to Ce(16) ⊂ Ce(9) [i] based on
(38, 38+17, large)-Net in Base 49 — Upper bound on s
There is no (38, 55, large)-net in base 49, because
- 15 times m-reduction [i] would yield (38, 40, large)-net in base 49, but