Best Known (27, 27+21, s)-Nets in Base 49
(27, 27+21, 344)-Net over F49 — Constructive and digital
Digital (27, 48, 344)-net over F49, using
- t-expansion [i] based on digital (21, 48, 344)-net over F49, using
- net from sequence [i] based on digital (21, 343)-sequence over F49, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F49 with g(F) = 21 and N(F) ≥ 344, using
- the Hermitian function field over F49 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F49 with g(F) = 21 and N(F) ≥ 344, using
- net from sequence [i] based on digital (21, 343)-sequence over F49, using
(27, 27+21, 2497)-Net over F49 — Digital
Digital (27, 48, 2497)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4948, 2497, F49, 21) (dual of [2497, 2449, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(4948, 2509, F49, 21) (dual of [2509, 2461, 22]-code), using
- 99 step Varšamov–Edel lengthening with (ri) = (4, 0, 0, 1, 8 times 0, 1, 23 times 0, 1, 62 times 0) [i] based on linear OA(4941, 2403, F49, 21) (dual of [2403, 2362, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(19) [i] based on
- linear OA(4941, 2401, F49, 21) (dual of [2401, 2360, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- 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(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(20) ⊂ Ce(19) [i] based on
- 99 step Varšamov–Edel lengthening with (ri) = (4, 0, 0, 1, 8 times 0, 1, 23 times 0, 1, 62 times 0) [i] based on linear OA(4941, 2403, F49, 21) (dual of [2403, 2362, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(4948, 2509, F49, 21) (dual of [2509, 2461, 22]-code), using
(27, 27+21, 8291944)-Net in Base 49 — Upper bound on s
There is no (27, 48, 8291945)-net in base 49, because
- 1 times m-reduction [i] would yield (27, 47, 8291945)-net in base 49, but
- the generalized Rao bound for nets shows that 49m ≥ 27 492607 678166 017508 455306 912840 143213 483429 761483 152006 028767 430205 769672 787425 > 4947 [i]