Best Known (71, 86, s)-Nets in Base 5
(71, 86, 11163)-Net over F5 — Constructive and digital
Digital (71, 86, 11163)-net over F5, using
- net defined by OOA [i] based on linear OOA(586, 11163, F5, 15, 15) (dual of [(11163, 15), 167359, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(586, 78142, F5, 15) (dual of [78142, 78056, 16]-code), using
- construction X4 applied to C([0,7]) ⊂ C([0,6]) [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(571, 78126, F5, 13) (dual of [78126, 78055, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 78126 | 514−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(515, 16, F5, 15) (dual of [16, 1, 16]-code or 16-arc in PG(14,5)), using
- dual of repetition code with length 16 [i]
- linear OA(51, 16, F5, 1) (dual of [16, 15, 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 X4 applied to C([0,7]) ⊂ C([0,6]) [i] based on
- OOA 7-folding and stacking with additional row [i] based on linear OA(586, 78142, F5, 15) (dual of [78142, 78056, 16]-code), using
(71, 86, 52658)-Net over F5 — Digital
Digital (71, 86, 52658)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(586, 52658, F5, 15) (dual of [52658, 52572, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(586, 78141, F5, 15) (dual of [78141, 78055, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,6]) [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(571, 78126, F5, 13) (dual of [78126, 78055, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 78126 | 514−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(51, 15, F5, 1) (dual of [15, 14, 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([0,7]) ⊂ C([0,6]) [i] based on
- discarding factors / shortening the dual code based on linear OA(586, 78141, F5, 15) (dual of [78141, 78055, 16]-code), using
(71, 86, large)-Net in Base 5 — Upper bound on s
There is no (71, 86, large)-net in base 5, because
- 13 times m-reduction [i] would yield (71, 73, large)-net in base 5, but