Best Known (87, 87+15, s)-Nets in Base 5
(87, 87+15, 55808)-Net over F5 — Constructive and digital
Digital (87, 102, 55808)-net over F5, using
- net defined by OOA [i] based on linear OOA(5102, 55808, F5, 15, 15) (dual of [(55808, 15), 837018, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(5102, 390657, F5, 15) (dual of [390657, 390555, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(5102, 390663, F5, 15) (dual of [390663, 390561, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,5]) [i] based on
- linear OA(597, 390626, F5, 15) (dual of [390626, 390529, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 390626 | 516−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- 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(55, 37, F5, 3) (dual of [37, 32, 4]-code or 37-cap in PG(4,5)), using
- discarding factors / shortening the dual code based on linear OA(55, 42, F5, 3) (dual of [42, 37, 4]-code or 42-cap in PG(4,5)), using
- construction X applied to C([0,7]) ⊂ C([0,5]) [i] based on
- discarding factors / shortening the dual code based on linear OA(5102, 390663, F5, 15) (dual of [390663, 390561, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(5102, 390657, F5, 15) (dual of [390657, 390555, 16]-code), using
(87, 87+15, 381764)-Net over F5 — Digital
Digital (87, 102, 381764)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5102, 381764, F5, 15) (dual of [381764, 381662, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(5102, 390663, F5, 15) (dual of [390663, 390561, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,5]) [i] based on
- linear OA(597, 390626, F5, 15) (dual of [390626, 390529, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 390626 | 516−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- 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(55, 37, F5, 3) (dual of [37, 32, 4]-code or 37-cap in PG(4,5)), using
- discarding factors / shortening the dual code based on linear OA(55, 42, F5, 3) (dual of [42, 37, 4]-code or 42-cap in PG(4,5)), using
- construction X applied to C([0,7]) ⊂ C([0,5]) [i] based on
- discarding factors / shortening the dual code based on linear OA(5102, 390663, F5, 15) (dual of [390663, 390561, 16]-code), using
(87, 87+15, large)-Net in Base 5 — Upper bound on s
There is no (87, 102, large)-net in base 5, because
- 13 times m-reduction [i] would yield (87, 89, large)-net in base 5, but