Best Known (146−23, 146, s)-Nets in Base 5
(146−23, 146, 35512)-Net over F5 — Constructive and digital
Digital (123, 146, 35512)-net over F5, using
- 51 times duplication [i] based on digital (122, 145, 35512)-net over F5, using
- net defined by OOA [i] based on linear OOA(5145, 35512, F5, 23, 23) (dual of [(35512, 23), 816631, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(5145, 390633, F5, 23) (dual of [390633, 390488, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(21) [i] based on
- linear OA(5145, 390625, F5, 23) (dual of [390625, 390480, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(5137, 390625, F5, 22) (dual of [390625, 390488, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(50, 8, F5, 0) (dual of [8, 8, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(50, s, F5, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(22) ⊂ Ce(21) [i] based on
- OOA 11-folding and stacking with additional row [i] based on linear OA(5145, 390633, F5, 23) (dual of [390633, 390488, 24]-code), using
- net defined by OOA [i] based on linear OOA(5145, 35512, F5, 23, 23) (dual of [(35512, 23), 816631, 24]-NRT-code), using
(146−23, 146, 195321)-Net over F5 — Digital
Digital (123, 146, 195321)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(5146, 195321, F5, 2, 23) (dual of [(195321, 2), 390496, 24]-NRT-code), using
- OOA 2-folding [i] based on linear OA(5146, 390642, F5, 23) (dual of [390642, 390496, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(5146, 390643, F5, 23) (dual of [390643, 390497, 24]-code), using
- construction X applied to C([0,11]) ⊂ C([0,10]) [i] based on
- linear OA(5145, 390626, F5, 23) (dual of [390626, 390481, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 390626 | 516−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- 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(51, 17, F5, 1) (dual of [17, 16, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(51, s, F5, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,11]) ⊂ C([0,10]) [i] based on
- discarding factors / shortening the dual code based on linear OA(5146, 390643, F5, 23) (dual of [390643, 390497, 24]-code), using
- OOA 2-folding [i] based on linear OA(5146, 390642, F5, 23) (dual of [390642, 390496, 24]-code), using
(146−23, 146, large)-Net in Base 5 — Upper bound on s
There is no (123, 146, large)-net in base 5, because
- 21 times m-reduction [i] would yield (123, 125, large)-net in base 5, but