Best Known (75, 75+15, s)-Nets in Base 5
(75, 75+15, 11165)-Net over F5 — Constructive and digital
Digital (75, 90, 11165)-net over F5, using
- net defined by OOA [i] based on linear OOA(590, 11165, F5, 15, 15) (dual of [(11165, 15), 167385, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(590, 78156, F5, 15) (dual of [78156, 78066, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(590, 78159, F5, 15) (dual of [78159, 78069, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,5]) [i] based on
- linear OA(585, 78126, F5, 15) (dual of [78126, 78041, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 78126 | 514−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(557, 78126, F5, 11) (dual of [78126, 78069, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 78126 | 514−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(55, 33, F5, 3) (dual of [33, 28, 4]-code or 33-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(590, 78159, F5, 15) (dual of [78159, 78069, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(590, 78156, F5, 15) (dual of [78156, 78066, 16]-code), using
(75, 75+15, 78159)-Net over F5 — Digital
Digital (75, 90, 78159)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(590, 78159, F5, 15) (dual of [78159, 78069, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,5]) [i] based on
- linear OA(585, 78126, F5, 15) (dual of [78126, 78041, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 78126 | 514−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(557, 78126, F5, 11) (dual of [78126, 78069, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 78126 | 514−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(55, 33, F5, 3) (dual of [33, 28, 4]-code or 33-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
(75, 75+15, large)-Net in Base 5 — Upper bound on s
There is no (75, 90, large)-net in base 5, because
- 13 times m-reduction [i] would yield (75, 77, large)-net in base 5, but