Best Known (18, 18+17, s)-Nets in Base 49
(18, 18+17, 301)-Net over F49 — Constructive and digital
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
(18, 18+17, 1204)-Net over F49 — Digital
Digital (18, 35, 1204)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4935, 1204, F49, 2, 17) (dual of [(1204, 2), 2373, 18]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4935, 2408, F49, 17) (dual of [2408, 2373, 18]-code), using
- discarding factors / shortening the dual code 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
- discarding factors / shortening the dual code based on linear OA(4935, 2409, F49, 17) (dual of [2409, 2374, 18]-code), using
- OOA 2-folding [i] based on linear OA(4935, 2408, F49, 17) (dual of [2408, 2373, 18]-code), using
(18, 18+17, 1196136)-Net in Base 49 — Upper bound on s
There is no (18, 35, 1196137)-net in base 49, because
- 1 times m-reduction [i] would yield (18, 34, 1196137)-net in base 49, but
- the generalized Rao bound for nets shows that 49m ≥ 2928 649662 073970 432696 131395 266521 728557 266010 290665 892225 > 4934 [i]