Best Known (41, 41+19, s)-Nets in Base 16
(41, 41+19, 1032)-Net over F16 — Constructive and digital
Digital (41, 60, 1032)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (11, 20, 516)-net over F16, using
- trace code for nets [i] based on digital (1, 10, 258)-net over F256, using
- net from sequence [i] based on digital (1, 257)-sequence over F256, using
- trace code for nets [i] based on digital (1, 10, 258)-net over F256, using
- digital (21, 40, 516)-net over F16, using
- trace code for nets [i] based on digital (1, 20, 258)-net over F256, using
- net from sequence [i] based on digital (1, 257)-sequence over F256 (see above)
- trace code for nets [i] based on digital (1, 20, 258)-net over F256, using
- digital (11, 20, 516)-net over F16, using
(41, 41+19, 5262)-Net over F16 — Digital
Digital (41, 60, 5262)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1660, 5262, F16, 19) (dual of [5262, 5202, 20]-code), using
- 1155 step Varšamov–Edel lengthening with (ri) = (3, 0, 0, 0, 1, 15 times 0, 1, 54 times 0, 1, 151 times 0, 1, 337 times 0, 1, 589 times 0) [i] based on linear OA(1652, 4099, F16, 19) (dual of [4099, 4047, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(17) [i] based on
- linear OA(1652, 4096, F16, 19) (dual of [4096, 4044, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(1649, 4096, F16, 18) (dual of [4096, 4047, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(160, 3, F16, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(160, s, F16, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(18) ⊂ Ce(17) [i] based on
- 1155 step Varšamov–Edel lengthening with (ri) = (3, 0, 0, 0, 1, 15 times 0, 1, 54 times 0, 1, 151 times 0, 1, 337 times 0, 1, 589 times 0) [i] based on linear OA(1652, 4099, F16, 19) (dual of [4099, 4047, 20]-code), using
(41, 41+19, large)-Net in Base 16 — Upper bound on s
There is no (41, 60, large)-net in base 16, because
- 17 times m-reduction [i] would yield (41, 43, large)-net in base 16, but