Best Known (140−34, 140, s)-Nets in Base 5
(140−34, 140, 460)-Net over F5 — Constructive and digital
Digital (106, 140, 460)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (35, 52, 208)-net over F5, using
- trace code for nets [i] based on digital (9, 26, 104)-net over F25, using
- net from sequence [i] based on digital (9, 103)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 9 and N(F) ≥ 104, using
- net from sequence [i] based on digital (9, 103)-sequence over F25, using
- trace code for nets [i] based on digital (9, 26, 104)-net over F25, using
- digital (54, 88, 252)-net over F5, using
- trace code for nets [i] based on digital (10, 44, 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, 44, 126)-net over F25, using
- digital (35, 52, 208)-net over F5, using
(140−34, 140, 3193)-Net over F5 — Digital
Digital (106, 140, 3193)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5140, 3193, F5, 34) (dual of [3193, 3053, 35]-code), using
- 59 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 0, 1, 14 times 0, 1, 39 times 0) [i] based on linear OA(5136, 3130, F5, 34) (dual of [3130, 2994, 35]-code), using
- construction X applied to Ce(33) ⊂ Ce(32) [i] based on
- linear OA(5136, 3125, F5, 34) (dual of [3125, 2989, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(5131, 3125, F5, 33) (dual of [3125, 2994, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [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(33) ⊂ Ce(32) [i] based on
- 59 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 0, 1, 14 times 0, 1, 39 times 0) [i] based on linear OA(5136, 3130, F5, 34) (dual of [3130, 2994, 35]-code), using
(140−34, 140, 1023537)-Net in Base 5 — Upper bound on s
There is no (106, 140, 1023538)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 71 747571 370521 409205 446279 234064 085732 440696 860402 926220 615187 592864 830673 864226 804097 684893 413065 > 5140 [i]