Best Known (39−20, 39, s)-Nets in Base 49
(39−20, 39, 240)-Net over F49 — Constructive and digital
Digital (19, 39, 240)-net over F49, using
- net defined by OOA [i] based on linear OOA(4939, 240, F49, 20, 20) (dual of [(240, 20), 4761, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(4939, 2400, F49, 20) (dual of [2400, 2361, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(4939, 2401, F49, 20) (dual of [2401, 2362, 21]-code), using
- an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- discarding factors / shortening the dual code based on linear OA(4939, 2401, F49, 20) (dual of [2401, 2362, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(4939, 2400, F49, 20) (dual of [2400, 2361, 21]-code), using
(39−20, 39, 801)-Net over F49 — Digital
Digital (19, 39, 801)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4939, 801, F49, 3, 20) (dual of [(801, 3), 2364, 21]-NRT-code), using
- OOA 3-folding [i] based on linear OA(4939, 2403, F49, 20) (dual of [2403, 2364, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(18) [i] based on
- linear OA(4939, 2401, F49, 20) (dual of [2401, 2362, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(4937, 2401, F49, 19) (dual of [2401, 2364, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [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(19) ⊂ Ce(18) [i] based on
- OOA 3-folding [i] based on linear OA(4939, 2403, F49, 20) (dual of [2403, 2364, 21]-code), using
(39−20, 39, 368548)-Net in Base 49 — Upper bound on s
There is no (19, 39, 368549)-net in base 49, because
- the generalized Rao bound for nets shows that 49m ≥ 827288 894417 077391 598169 210372 969560 658231 795620 682608 892631 138401 > 4939 [i]