Best Known (84−19, 84, s)-Nets in Base 5
(84−19, 84, 384)-Net over F5 — Constructive and digital
Digital (65, 84, 384)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (17, 26, 132)-net over F5, using
- trace code for nets [i] based on digital (4, 13, 66)-net over F25, using
- net from sequence [i] based on digital (4, 65)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 4 and N(F) ≥ 66, using
- net from sequence [i] based on digital (4, 65)-sequence over F25, using
- trace code for nets [i] based on digital (4, 13, 66)-net over F25, using
- digital (39, 58, 252)-net over F5, using
- trace code for nets [i] based on digital (10, 29, 126)-net over F25, using
- net from sequence [i] based on digital (10, 125)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 10 and N(F) ≥ 126, using
- the Hermitian function field over F25 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 10 and N(F) ≥ 126, using
- net from sequence [i] based on digital (10, 125)-sequence over F25, using
- trace code for nets [i] based on digital (10, 29, 126)-net over F25, using
- digital (17, 26, 132)-net over F5, using
(84−19, 84, 3546)-Net over F5 — Digital
Digital (65, 84, 3546)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(584, 3546, F5, 19) (dual of [3546, 3462, 20]-code), using
- 408 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 4 times 0, 1, 12 times 0, 1, 27 times 0, 1, 57 times 0, 1, 112 times 0, 1, 188 times 0) [i] based on linear OA(576, 3130, F5, 19) (dual of [3130, 3054, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(17) [i] based on
- linear OA(576, 3125, F5, 19) (dual of [3125, 3049, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(571, 3125, F5, 18) (dual of [3125, 3054, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(50, 5, F5, 0) (dual of [5, 5, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(50, s, F5, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(18) ⊂ Ce(17) [i] based on
- 408 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 4 times 0, 1, 12 times 0, 1, 27 times 0, 1, 57 times 0, 1, 112 times 0, 1, 188 times 0) [i] based on linear OA(576, 3130, F5, 19) (dual of [3130, 3054, 20]-code), using
(84−19, 84, 2895657)-Net in Base 5 — Upper bound on s
There is no (65, 84, 2895658)-net in base 5, because
- 1 times m-reduction [i] would yield (65, 83, 2895658)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 10339 766423 896687 670424 105760 429042 148331 433221 852925 888425 > 583 [i]