Best Known (23, 23+16, s)-Nets in Base 49
(23, 23+16, 350)-Net over F49 — Constructive and digital
Digital (23, 39, 350)-net over F49, using
- (u, u+v)-construction [i] based on
- digital (0, 8, 50)-net over F49, using
- net from sequence [i] based on digital (0, 49)-sequence over F49, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F49 with g(F) = 0 and N(F) ≥ 50, using
- the rational function field F49(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 49)-sequence over F49, using
- digital (15, 31, 300)-net over F49, using
- net defined by OOA [i] based on linear OOA(4931, 300, F49, 16, 16) (dual of [(300, 16), 4769, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(4931, 2400, F49, 16) (dual of [2400, 2369, 17]-code), using
- discarding factors / shortening the dual code 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]
- discarding factors / shortening the dual code based on linear OA(4931, 2401, F49, 16) (dual of [2401, 2370, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(4931, 2400, F49, 16) (dual of [2400, 2369, 17]-code), using
- net defined by OOA [i] based on linear OOA(4931, 300, F49, 16, 16) (dual of [(300, 16), 4769, 17]-NRT-code), using
- digital (0, 8, 50)-net over F49, using
(23, 23+16, 3364)-Net over F49 — Digital
Digital (23, 39, 3364)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4939, 3364, F49, 16) (dual of [3364, 3325, 17]-code), using
- 953 step Varšamov–Edel lengthening with (ri) = (3, 1, 4 times 0, 1, 21 times 0, 1, 79 times 0, 1, 252 times 0, 1, 591 times 0) [i] based on linear OA(4931, 2403, F49, 16) (dual of [2403, 2372, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(14) [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(4929, 2401, F49, 15) (dual of [2401, 2372, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [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(15) ⊂ Ce(14) [i] based on
- 953 step Varšamov–Edel lengthening with (ri) = (3, 1, 4 times 0, 1, 21 times 0, 1, 79 times 0, 1, 252 times 0, 1, 591 times 0) [i] based on linear OA(4931, 2403, F49, 16) (dual of [2403, 2372, 17]-code), using
(23, 23+16, large)-Net in Base 49 — Upper bound on s
There is no (23, 39, large)-net in base 49, because
- 14 times m-reduction [i] would yield (23, 25, large)-net in base 49, but