Best Known (138−20, 138, s)-Nets in Base 5
(138−20, 138, 39068)-Net over F5 — Constructive and digital
Digital (118, 138, 39068)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (0, 10, 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 (108, 128, 39062)-net over F5, using
- net defined by OOA [i] based on linear OOA(5128, 39062, F5, 20, 20) (dual of [(39062, 20), 781112, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(5128, 390620, F5, 20) (dual of [390620, 390492, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(5128, 390625, F5, 20) (dual of [390625, 390497, 21]-code), using
- 1 times truncation [i] based on linear OA(5129, 390626, F5, 21) (dual of [390626, 390497, 22]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 390626 | 516−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- 1 times truncation [i] based on linear OA(5129, 390626, F5, 21) (dual of [390626, 390497, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(5128, 390625, F5, 20) (dual of [390625, 390497, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(5128, 390620, F5, 20) (dual of [390620, 390492, 21]-code), using
- net defined by OOA [i] based on linear OOA(5128, 39062, F5, 20, 20) (dual of [(39062, 20), 781112, 21]-NRT-code), using
- digital (0, 10, 6)-net over F5, using
(138−20, 138, 390674)-Net over F5 — Digital
Digital (118, 138, 390674)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5138, 390674, F5, 20) (dual of [390674, 390536, 21]-code), using
- construction X applied to Ce(20) ⊂ Ce(13) [i] based on
- linear OA(5129, 390625, F5, 21) (dual of [390625, 390496, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(589, 390625, F5, 14) (dual of [390625, 390536, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(59, 49, F5, 5) (dual of [49, 40, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(59, 62, F5, 5) (dual of [62, 53, 6]-code), using
- a “GraCyc†code from Grassl’s database [i]
- discarding factors / shortening the dual code based on linear OA(59, 62, F5, 5) (dual of [62, 53, 6]-code), using
- construction X applied to Ce(20) ⊂ Ce(13) [i] based on
(138−20, 138, large)-Net in Base 5 — Upper bound on s
There is no (118, 138, large)-net in base 5, because
- 18 times m-reduction [i] would yield (118, 120, large)-net in base 5, but