Best Known (53−10, 53, s)-Nets in Base 5
(53−10, 53, 3131)-Net over F5 — Constructive and digital
Digital (43, 53, 3131)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (0, 5, 6)-net over F5, using
- net from sequence [i] based on digital (0, 5)-sequence over F5, using
- Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 0 and N(F) ≥ 6, using
- the rational function field F5(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 5)-sequence over F5, using
- digital (38, 48, 3125)-net over F5, using
- net defined by OOA [i] based on linear OOA(548, 3125, F5, 10, 10) (dual of [(3125, 10), 31202, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(548, 15625, F5, 10) (dual of [15625, 15577, 11]-code), using
- 1 times truncation [i] based on linear OA(549, 15626, F5, 11) (dual of [15626, 15577, 12]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 15626 | 512−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- 1 times truncation [i] based on linear OA(549, 15626, F5, 11) (dual of [15626, 15577, 12]-code), using
- OA 5-folding and stacking [i] based on linear OA(548, 15625, F5, 10) (dual of [15625, 15577, 11]-code), using
- net defined by OOA [i] based on linear OOA(548, 3125, F5, 10, 10) (dual of [(3125, 10), 31202, 11]-NRT-code), using
- digital (0, 5, 6)-net over F5, using
(53−10, 53, 15651)-Net over F5 — Digital
Digital (43, 53, 15651)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(553, 15651, F5, 10) (dual of [15651, 15598, 11]-code), using
- construction X applied to Ce(10) ⊂ Ce(5) [i] based on
- linear OA(549, 15625, F5, 11) (dual of [15625, 15576, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(525, 15625, F5, 6) (dual of [15625, 15600, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(54, 26, F5, 3) (dual of [26, 22, 4]-code or 26-cap in PG(3,5)), using
- construction X applied to Ce(10) ⊂ Ce(5) [i] based on
(53−10, 53, large)-Net in Base 5 — Upper bound on s
There is no (43, 53, large)-net in base 5, because
- 8 times m-reduction [i] would yield (43, 45, large)-net in base 5, but