Best Known (123, 123+21, s)-Nets in Base 5
(123, 123+21, 39082)-Net over F5 — Constructive and digital
Digital (123, 144, 39082)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (5, 15, 20)-net over F5, using
- net from sequence [i] based on digital (5, 19)-sequence over F5, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 5 and N(F) ≥ 20, using
- net from sequence [i] based on digital (5, 19)-sequence over F5, using
- digital (108, 129, 39062)-net over F5, using
- net defined by OOA [i] based on linear OOA(5129, 39062, F5, 21, 21) (dual of [(39062, 21), 820173, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(5129, 390621, F5, 21) (dual of [390621, 390492, 22]-code), using
- discarding factors / shortening the dual code 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]
- discarding factors / shortening the dual code based on linear OA(5129, 390625, F5, 21) (dual of [390625, 390496, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(5129, 390621, F5, 21) (dual of [390621, 390492, 22]-code), using
- net defined by OOA [i] based on linear OOA(5129, 39062, F5, 21, 21) (dual of [(39062, 21), 820173, 22]-NRT-code), using
- digital (5, 15, 20)-net over F5, using
(123, 123+21, 361251)-Net over F5 — Digital
Digital (123, 144, 361251)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5144, 361251, F5, 21) (dual of [361251, 361107, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(5144, 390688, F5, 21) (dual of [390688, 390544, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,6]) [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]
- linear OA(581, 390626, F5, 13) (dual of [390626, 390545, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 390626 | 516−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(515, 62, F5, 7) (dual of [62, 47, 8]-code), using
- construction X applied to C([0,10]) ⊂ C([0,6]) [i] based on
- discarding factors / shortening the dual code based on linear OA(5144, 390688, F5, 21) (dual of [390688, 390544, 22]-code), using
(123, 123+21, large)-Net in Base 5 — Upper bound on s
There is no (123, 144, large)-net in base 5, because
- 19 times m-reduction [i] would yield (123, 125, large)-net in base 5, but