Best Known (16, 33, s)-Nets in Base 49
(16, 33, 300)-Net over F49 — Constructive and digital
Digital (16, 33, 300)-net over F49, using
- net defined by OOA [i] based on linear OOA(4933, 300, F49, 17, 17) (dual of [(300, 17), 5067, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [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]
- OOA 8-folding and stacking with additional row [i] based on linear OA(4933, 2401, F49, 17) (dual of [2401, 2368, 18]-code), using
(16, 33, 801)-Net over F49 — Digital
Digital (16, 33, 801)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4933, 801, F49, 3, 17) (dual of [(801, 3), 2370, 18]-NRT-code), using
- OOA 3-folding [i] based on linear OA(4933, 2403, F49, 17) (dual of [2403, 2370, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(15) [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(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(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(16) ⊂ Ce(15) [i] based on
- OOA 3-folding [i] based on linear OA(4933, 2403, F49, 17) (dual of [2403, 2370, 18]-code), using
(16, 33, 452094)-Net in Base 49 — Upper bound on s
There is no (16, 33, 452095)-net in base 49, because
- 1 times m-reduction [i] would yield (16, 32, 452095)-net in base 49, but
- the generalized Rao bound for nets shows that 49m ≥ 1 219762 241016 352728 453017 937439 237825 200328 877157 973633 > 4932 [i]