Best Known (69, 69+23, s)-Nets in Base 5
(69, 69+23, 312)-Net over F5 — Constructive and digital
Digital (69, 92, 312)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (17, 28, 104)-net over F5, using
- trace code for nets [i] based on digital (3, 14, 52)-net over F25, using
- net from sequence [i] based on digital (3, 51)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 3 and N(F) ≥ 52, using
- net from sequence [i] based on digital (3, 51)-sequence over F25, using
- trace code for nets [i] based on digital (3, 14, 52)-net over F25, using
- digital (41, 64, 208)-net over F5, using
- trace code for nets [i] based on digital (9, 32, 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, 32, 104)-net over F25, using
- digital (17, 28, 104)-net over F5, using
(69, 69+23, 2305)-Net over F5 — Digital
Digital (69, 92, 2305)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(592, 2305, F5, 23) (dual of [2305, 2213, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(592, 3137, F5, 23) (dual of [3137, 3045, 24]-code), using
- construction X applied to C([0,11]) ⊂ C([0,10]) [i] based on
- linear OA(591, 3126, F5, 23) (dual of [3126, 3035, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 3126 | 510−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- linear OA(581, 3126, F5, 21) (dual of [3126, 3045, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 3126 | 510−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(51, 11, F5, 1) (dual of [11, 10, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(51, s, F5, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,11]) ⊂ C([0,10]) [i] based on
- discarding factors / shortening the dual code based on linear OA(592, 3137, F5, 23) (dual of [3137, 3045, 24]-code), using
(69, 69+23, 743597)-Net in Base 5 — Upper bound on s
There is no (69, 92, 743598)-net in base 5, because
- 1 times m-reduction [i] would yield (69, 91, 743598)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 4039 004332 895292 219553 883758 095052 485822 130150 479971 387219 323353 > 591 [i]