Best Known (10, 10+9, s)-Nets in Base 49
(10, 10+9, 602)-Net over F49 — Constructive and digital
Digital (10, 19, 602)-net over F49, using
- net defined by OOA [i] based on linear OOA(4919, 602, F49, 9, 9) (dual of [(602, 9), 5399, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(4919, 2409, F49, 9) (dual of [2409, 2390, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(5) [i] based on
- linear OA(4917, 2401, F49, 9) (dual of [2401, 2384, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(4911, 2401, F49, 6) (dual of [2401, 2390, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [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(8) ⊂ Ce(5) [i] based on
- OOA 4-folding and stacking with additional row [i] based on linear OA(4919, 2409, F49, 9) (dual of [2409, 2390, 10]-code), using
(10, 10+9, 1560)-Net over F49 — Digital
Digital (10, 19, 1560)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4919, 1560, F49, 9) (dual of [1560, 1541, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(4919, 2409, F49, 9) (dual of [2409, 2390, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(5) [i] based on
- linear OA(4917, 2401, F49, 9) (dual of [2401, 2384, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(4911, 2401, F49, 6) (dual of [2401, 2390, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [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(8) ⊂ Ce(5) [i] based on
- discarding factors / shortening the dual code based on linear OA(4919, 2409, F49, 9) (dual of [2409, 2390, 10]-code), using
(10, 10+9, 1860773)-Net in Base 49 — Upper bound on s
There is no (10, 19, 1860774)-net in base 49, because
- 1 times m-reduction [i] would yield (10, 18, 1860774)-net in base 49, but
- the generalized Rao bound for nets shows that 49m ≥ 2 651732 446305 601608 442907 937409 > 4918 [i]