Best Known (87−22, 87, s)-Nets in Base 5
(87−22, 87, 304)-Net over F5 — Constructive and digital
Digital (65, 87, 304)-net over F5, using
- 1 times m-reduction [i] based on digital (65, 88, 304)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (11, 22, 52)-net over F5, using
- trace code for nets [i] based on digital (0, 11, 26)-net over F25, using
- net from sequence [i] based on digital (0, 25)-sequence over F25, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 0 and N(F) ≥ 26, using
- the rational function field F25(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 25)-sequence over F25, using
- trace code for nets [i] based on digital (0, 11, 26)-net over F25, using
- digital (43, 66, 252)-net over F5, using
- trace code for nets [i] based on digital (10, 33, 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, 33, 126)-net over F25, using
- digital (11, 22, 52)-net over F5, using
- (u, u+v)-construction [i] based on
(87−22, 87, 2089)-Net over F5 — Digital
Digital (65, 87, 2089)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(587, 2089, F5, 22) (dual of [2089, 2002, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(587, 3131, F5, 22) (dual of [3131, 3044, 23]-code), using
- 1 times code embedding in larger space [i] based on linear OA(586, 3130, F5, 22) (dual of [3130, 3044, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(20) [i] based on
- linear OA(586, 3125, F5, 22) (dual of [3125, 3039, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(581, 3125, F5, 21) (dual of [3125, 3044, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [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(21) ⊂ Ce(20) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(586, 3130, F5, 22) (dual of [3130, 3044, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(587, 3131, F5, 22) (dual of [3131, 3044, 23]-code), using
(87−22, 87, 414155)-Net in Base 5 — Upper bound on s
There is no (65, 87, 414156)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 6 462448 763022 918119 911734 096717 731723 018768 000125 070153 791025 > 587 [i]