Best Known (40−16, 40, s)-Nets in Base 49
(40−16, 40, 351)-Net over F49 — Constructive and digital
Digital (24, 40, 351)-net over F49, using
- (u, u+v)-construction [i] based on
- digital (1, 9, 51)-net over F49, using
- net from sequence [i] based on digital (1, 50)-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 (1, 9, 51)-net over F49, using
(40−16, 40, 4313)-Net over F49 — Digital
Digital (24, 40, 4313)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4940, 4313, F49, 16) (dual of [4313, 4273, 17]-code), using
- 1901 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, 1, 947 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
- 1901 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, 1, 947 times 0) [i] based on linear OA(4931, 2403, F49, 16) (dual of [2403, 2372, 17]-code), using
(40−16, 40, large)-Net in Base 49 — Upper bound on s
There is no (24, 40, large)-net in base 49, because
- 14 times m-reduction [i] would yield (24, 26, large)-net in base 49, but