Best Known (20, 20+17, s)-Nets in Base 49
(20, 20+17, 301)-Net over F49 — Constructive and digital
Digital (20, 37, 301)-net over F49, using
- 492 times duplication [i] based on digital (18, 35, 301)-net over F49, using
- net defined by OOA [i] based on linear OOA(4935, 301, F49, 17, 17) (dual of [(301, 17), 5082, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(4935, 2409, F49, 17) (dual of [2409, 2374, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(13) [i] based on
- linear OA(4933, 2401, F49, 17) (dual of [2401, 2368, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(4927, 2401, F49, 14) (dual of [2401, 2374, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(492, 8, F49, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,49)), using
- discarding factors / shortening the dual code based on linear OA(492, 49, F49, 2) (dual of [49, 47, 3]-code or 49-arc in PG(1,49)), using
- Reed–Solomon code RS(47,49) [i]
- discarding factors / shortening the dual code based on linear OA(492, 49, F49, 2) (dual of [49, 47, 3]-code or 49-arc in PG(1,49)), using
- construction X applied to Ce(16) ⊂ Ce(13) [i] based on
- OOA 8-folding and stacking with additional row [i] based on linear OA(4935, 2409, F49, 17) (dual of [2409, 2374, 18]-code), using
- net defined by OOA [i] based on linear OOA(4935, 301, F49, 17, 17) (dual of [(301, 17), 5082, 18]-NRT-code), using
(20, 20+17, 1517)-Net over F49 — Digital
Digital (20, 37, 1517)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4937, 1517, F49, 17) (dual of [1517, 1480, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(4937, 2415, F49, 17) (dual of [2415, 2378, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(11) [i] based on
- linear OA(4933, 2401, F49, 17) (dual of [2401, 2368, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(4923, 2401, F49, 12) (dual of [2401, 2378, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(494, 14, F49, 4) (dual of [14, 10, 5]-code or 14-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(16) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(4937, 2415, F49, 17) (dual of [2415, 2378, 18]-code), using
(20, 20+17, 3164686)-Net in Base 49 — Upper bound on s
There is no (20, 37, 3164687)-net in base 49, because
- 1 times m-reduction [i] would yield (20, 36, 3164687)-net in base 49, but
- the generalized Rao bound for nets shows that 49m ≥ 7 031686 587641 570140 223766 296123 070372 980013 421723 597361 460865 > 4936 [i]