Best Known (58, 69, s)-Nets in Base 5
(58, 69, 78130)-Net over F5 — Constructive and digital
Digital (58, 69, 78130)-net over F5, using
- net defined by OOA [i] based on linear OOA(569, 78130, F5, 11, 11) (dual of [(78130, 11), 859361, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(569, 390651, F5, 11) (dual of [390651, 390582, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(6) [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(54, 26, F5, 3) (dual of [26, 22, 4]-code or 26-cap in PG(3,5)), using
- construction X applied to Ce(10) ⊂ Ce(6) [i] based on
- OOA 5-folding and stacking with additional row [i] based on linear OA(569, 390651, F5, 11) (dual of [390651, 390582, 12]-code), using
(58, 69, 198055)-Net over F5 — Digital
Digital (58, 69, 198055)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(569, 198055, F5, 11) (dual of [198055, 197986, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(569, 390646, F5, 11) (dual of [390646, 390577, 12]-code), using
- construction X applied to C([0,5]) ⊂ C([0,3]) [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]
- linear OA(549, 390626, F5, 7) (dual of [390626, 390577, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 390626 | 516−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- linear OA(54, 20, F5, 3) (dual of [20, 16, 4]-code or 20-cap in PG(3,5)), using
- construction X applied to C([0,5]) ⊂ C([0,3]) [i] based on
- discarding factors / shortening the dual code based on linear OA(569, 390646, F5, 11) (dual of [390646, 390577, 12]-code), using
(58, 69, large)-Net in Base 5 — Upper bound on s
There is no (58, 69, large)-net in base 5, because
- 9 times m-reduction [i] would yield (58, 60, large)-net in base 5, but