Best Known (69−10, 69, s)-Nets in Base 5
(69−10, 69, 78131)-Net over F5 — Constructive and digital
Digital (59, 69, 78131)-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 (54, 64, 78125)-net over F5, using
- net defined by OOA [i] based on linear OOA(564, 78125, F5, 10, 10) (dual of [(78125, 10), 781186, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(564, 390625, F5, 10) (dual of [390625, 390561, 11]-code), using
- 1 times truncation [i] based on linear OA(565, 390626, F5, 11) (dual of [390626, 390561, 12]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 390626 | 516−1, 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(565, 390626, F5, 11) (dual of [390626, 390561, 12]-code), using
- OA 5-folding and stacking [i] based on linear OA(564, 390625, F5, 10) (dual of [390625, 390561, 11]-code), using
- net defined by OOA [i] based on linear OOA(564, 78125, F5, 10, 10) (dual of [(78125, 10), 781186, 11]-NRT-code), using
- digital (0, 5, 6)-net over F5, using
(69−10, 69, 390654)-Net over F5 — Digital
Digital (59, 69, 390654)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(569, 390654, F5, 10) (dual of [390654, 390585, 11]-code), using
- construction XX applied to Ce(10) ⊂ Ce(6) ⊂ Ce(5) [i] based on
- linear OA(565, 390625, F5, 11) (dual of [390625, 390560, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(541, 390625, F5, 7) (dual of [390625, 390584, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(533, 390625, F5, 6) (dual of [390625, 390592, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(53, 28, F5, 2) (dual of [28, 25, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(53, 31, F5, 2) (dual of [31, 28, 3]-code), using
- Hamming code H(3,5) [i]
- discarding factors / shortening the dual code based on linear OA(53, 31, F5, 2) (dual of [31, 28, 3]-code), using
- linear OA(50, 1, F5, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(10) ⊂ Ce(6) ⊂ Ce(5) [i] based on
(69−10, 69, large)-Net in Base 5 — Upper bound on s
There is no (59, 69, large)-net in base 5, because
- 8 times m-reduction [i] would yield (59, 61, large)-net in base 5, but