Best Known (67, 67+21, s)-Nets in Base 5
(67, 67+21, 356)-Net over F5 — Constructive and digital
Digital (67, 88, 356)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (16, 26, 104)-net over F5, using
- trace code for nets [i] based on digital (3, 13, 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, 13, 52)-net over F25, using
- digital (41, 62, 252)-net over F5, using
- trace code for nets [i] based on digital (10, 31, 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, 31, 126)-net over F25, using
- digital (16, 26, 104)-net over F5, using
(67, 67+21, 3133)-Net over F5 — Digital
Digital (67, 88, 3133)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(588, 3133, F5, 21) (dual of [3133, 3045, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(588, 3147, F5, 21) (dual of [3147, 3059, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,7]) [i] based on
- 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(561, 3126, F5, 15) (dual of [3126, 3065, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 3126 | 510−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(57, 21, F5, 5) (dual of [21, 14, 6]-code), using
- construction X applied to C([0,10]) ⊂ C([0,7]) [i] based on
- discarding factors / shortening the dual code based on linear OA(588, 3147, F5, 21) (dual of [3147, 3059, 22]-code), using
(67, 67+21, 1364435)-Net in Base 5 — Upper bound on s
There is no (67, 88, 1364436)-net in base 5, because
- 1 times m-reduction [i] would yield (67, 87, 1364436)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 6 462355 666159 921034 038031 748921 151360 275071 927716 785299 272641 > 587 [i]