Best Known (27, 27+12, s)-Nets in Base 49
(27, 27+12, 19612)-Net over F49 — Constructive and digital
Digital (27, 39, 19612)-net over F49, using
- net defined by OOA [i] based on linear OOA(4939, 19612, F49, 12, 12) (dual of [(19612, 12), 235305, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(4939, 117672, F49, 12) (dual of [117672, 117633, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(5) [i] based on
- linear OA(4934, 117649, F49, 12) (dual of [117649, 117615, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 117648 = 493−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [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(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 Ce(11) ⊂ Ce(5) [i] based on
- OA 6-folding and stacking [i] based on linear OA(4939, 117672, F49, 12) (dual of [117672, 117633, 13]-code), using
(27, 27+12, 117672)-Net over F49 — Digital
Digital (27, 39, 117672)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4939, 117672, F49, 12) (dual of [117672, 117633, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(5) [i] based on
- linear OA(4934, 117649, F49, 12) (dual of [117649, 117615, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 117648 = 493−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [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(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 Ce(11) ⊂ Ce(5) [i] based on
(27, 27+12, large)-Net in Base 49 — Upper bound on s
There is no (27, 39, large)-net in base 49, because
- 10 times m-reduction [i] would yield (27, 29, large)-net in base 49, but