Best Known (39−17, 39, s)-Nets in Base 49
(39−17, 39, 344)-Net over F49 — Constructive and digital
Digital (22, 39, 344)-net over F49, using
- t-expansion [i] based on digital (21, 39, 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
(39−17, 39, 2490)-Net over F49 — Digital
Digital (22, 39, 2490)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4939, 2490, F49, 17) (dual of [2490, 2451, 18]-code), using
- 81 step Varšamov–Edel lengthening with (ri) = (4, 0, 0, 0, 1, 16 times 0, 1, 59 times 0) [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
- 81 step Varšamov–Edel lengthening with (ri) = (4, 0, 0, 0, 1, 16 times 0, 1, 59 times 0) [i] based on linear OA(4933, 2403, F49, 17) (dual of [2403, 2370, 18]-code), using
(39−17, 39, 8372979)-Net in Base 49 — Upper bound on s
There is no (22, 39, 8372980)-net in base 49, because
- 1 times m-reduction [i] would yield (22, 38, 8372980)-net in base 49, but
- the generalized Rao bound for nets shows that 49m ≥ 16883 066369 872946 093533 687374 270529 584848 619002 907500 763638 097409 > 4938 [i]