Best Known (39−12, 39, s)-Nets in Base 5
(39−12, 39, 142)-Net over F5 — Constructive and digital
Digital (27, 39, 142)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (1, 7, 10)-net over F5, using
- net from sequence [i] based on digital (1, 9)-sequence over F5, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 1 and N(F) ≥ 10, using
- net from sequence [i] based on digital (1, 9)-sequence over F5, using
- digital (20, 32, 132)-net over F5, using
- trace code for nets [i] based on digital (4, 16, 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, 16, 66)-net over F25, using
- digital (1, 7, 10)-net over F5, using
(39−12, 39, 507)-Net over F5 — Digital
Digital (27, 39, 507)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(539, 507, F5, 12) (dual of [507, 468, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(539, 632, F5, 12) (dual of [632, 593, 13]-code), using
- construction XX applied to Ce(11) ⊂ Ce(10) ⊂ Ce(8) [i] based on
- linear OA(537, 625, F5, 12) (dual of [625, 588, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(533, 625, F5, 11) (dual of [625, 592, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(529, 625, F5, 9) (dual of [625, 596, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [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
- linear OA(51, 2, F5, 1) (dual of [2, 1, 2]-code), using
- dual of repetition code with length 2 [i]
- construction XX applied to Ce(11) ⊂ Ce(10) ⊂ Ce(8) [i] based on
- discarding factors / shortening the dual code based on linear OA(539, 632, F5, 12) (dual of [632, 593, 13]-code), using
(39−12, 39, 26145)-Net in Base 5 — Upper bound on s
There is no (27, 39, 26146)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 1819 104245 094408 293002 942465 > 539 [i]