Best Known (36−18, 36, s)-Nets in Base 49
(36−18, 36, 267)-Net over F49 — Constructive and digital
Digital (18, 36, 267)-net over F49, using
- 491 times duplication [i] based on digital (17, 35, 267)-net over F49, using
- net defined by OOA [i] based on linear OOA(4935, 267, F49, 18, 18) (dual of [(267, 18), 4771, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(4935, 2403, F49, 18) (dual of [2403, 2368, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(16) [i] based on
- linear OA(4935, 2401, F49, 18) (dual of [2401, 2366, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- 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(490, 2, F49, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(490, s, F49, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(17) ⊂ Ce(16) [i] based on
- OA 9-folding and stacking [i] based on linear OA(4935, 2403, F49, 18) (dual of [2403, 2368, 19]-code), using
- net defined by OOA [i] based on linear OOA(4935, 267, F49, 18, 18) (dual of [(267, 18), 4771, 19]-NRT-code), using
(36−18, 36, 900)-Net over F49 — Digital
Digital (18, 36, 900)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4936, 900, F49, 2, 18) (dual of [(900, 2), 1764, 19]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4936, 1203, F49, 2, 18) (dual of [(1203, 2), 2370, 19]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4936, 2406, F49, 18) (dual of [2406, 2370, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(15) [i] based on
- linear OA(4935, 2401, F49, 18) (dual of [2401, 2366, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(4931, 2401, F49, 16) (dual of [2401, 2370, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(491, 5, F49, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(491, s, F49, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(17) ⊂ Ce(15) [i] based on
- OOA 2-folding [i] based on linear OA(4936, 2406, F49, 18) (dual of [2406, 2370, 19]-code), using
- discarding factors / shortening the dual code based on linear OOA(4936, 1203, F49, 2, 18) (dual of [(1203, 2), 2370, 19]-NRT-code), using
(36−18, 36, 498070)-Net in Base 49 — Upper bound on s
There is no (18, 36, 498071)-net in base 49, because
- the generalized Rao bound for nets shows that 49m ≥ 7 031731 309964 687006 162798 205658 381767 521867 889922 667432 279761 > 4936 [i]