Best Known (51−22, 51, s)-Nets in Base 49
(51−22, 51, 344)-Net over F49 — Constructive and digital
Digital (29, 51, 344)-net over F49, using
- t-expansion [i] based on digital (21, 51, 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
(51−22, 51, 2609)-Net over F49 — Digital
Digital (29, 51, 2609)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4951, 2609, F49, 22) (dual of [2609, 2558, 23]-code), using
- 198 step Varšamov–Edel lengthening with (ri) = (4, 0, 1, 6 times 0, 1, 17 times 0, 1, 48 times 0, 1, 121 times 0) [i] based on linear OA(4943, 2403, F49, 22) (dual of [2403, 2360, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(20) [i] based on
- linear OA(4943, 2401, F49, 22) (dual of [2401, 2358, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- 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(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(21) ⊂ Ce(20) [i] based on
- 198 step Varšamov–Edel lengthening with (ri) = (4, 0, 1, 6 times 0, 1, 17 times 0, 1, 48 times 0, 1, 121 times 0) [i] based on linear OA(4943, 2403, F49, 22) (dual of [2403, 2360, 23]-code), using
(51−22, 51, 7016753)-Net in Base 49 — Upper bound on s
There is no (29, 51, 7016754)-net in base 49, because
- the generalized Rao bound for nets shows that 49m ≥ 158 489397 000239 411941 771911 617104 432212 226809 026729 106759 294159 700253 407448 581956 766753 > 4951 [i]