Best Known (45−10, 45, s)-Nets in Base 5
(45−10, 45, 631)-Net over F5 — Constructive and digital
Digital (35, 45, 631)-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 (30, 40, 625)-net over F5, using
- net defined by OOA [i] based on linear OOA(540, 625, F5, 10, 10) (dual of [(625, 10), 6210, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(540, 3125, F5, 10) (dual of [3125, 3085, 11]-code), using
- 1 times truncation [i] based on linear OA(541, 3126, F5, 11) (dual of [3126, 3085, 12]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 3126 | 510−1, 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(541, 3126, F5, 11) (dual of [3126, 3085, 12]-code), using
- OA 5-folding and stacking [i] based on linear OA(540, 3125, F5, 10) (dual of [3125, 3085, 11]-code), using
- net defined by OOA [i] based on linear OOA(540, 625, F5, 10, 10) (dual of [(625, 10), 6210, 11]-NRT-code), using
- digital (0, 5, 6)-net over F5, using
(45−10, 45, 3506)-Net over F5 — Digital
Digital (35, 45, 3506)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(545, 3506, F5, 10) (dual of [3506, 3461, 11]-code), using
- 376 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 0, 1, 22 times 0, 1, 91 times 0, 1, 256 times 0) [i] based on linear OA(540, 3125, F5, 10) (dual of [3125, 3085, 11]-code), using
- 1 times truncation [i] based on linear OA(541, 3126, F5, 11) (dual of [3126, 3085, 12]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 3126 | 510−1, 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(541, 3126, F5, 11) (dual of [3126, 3085, 12]-code), using
- 376 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 0, 1, 22 times 0, 1, 91 times 0, 1, 256 times 0) [i] based on linear OA(540, 3125, F5, 10) (dual of [3125, 3085, 11]-code), using
(45−10, 45, 1272052)-Net in Base 5 — Upper bound on s
There is no (35, 45, 1272053)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 28 421715 719645 270388 665623 765765 > 545 [i]