Best Known (37−16, 37, s)-Nets in Base 49
(37−16, 37, 344)-Net over F49 — Constructive and digital
Digital (21, 37, 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
(37−16, 37, 2518)-Net over F49 — Digital
Digital (21, 37, 2518)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4937, 2518, F49, 16) (dual of [2518, 2481, 17]-code), using
- 107 step Varšamov–Edel lengthening with (ri) = (3, 4 times 0, 1, 21 times 0, 1, 79 times 0) [i] based on linear OA(4932, 2406, F49, 16) (dual of [2406, 2374, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(13) [i] based on
- 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(4927, 2401, F49, 14) (dual of [2401, 2374, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(491, 5, F49, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(491, s, F49, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(15) ⊂ Ce(13) [i] based on
- 107 step Varšamov–Edel lengthening with (ri) = (3, 4 times 0, 1, 21 times 0, 1, 79 times 0) [i] based on linear OA(4932, 2406, F49, 16) (dual of [2406, 2374, 17]-code), using
(37−16, 37, 5147607)-Net in Base 49 — Upper bound on s
There is no (21, 37, 5147608)-net in base 49, because
- the generalized Rao bound for nets shows that 49m ≥ 344 552644 531496 807879 594024 131711 258471 494397 252227 941474 079745 > 4937 [i]