Best Known (84, 84+15, s)-Nets in Base 5
(84, 84+15, 55806)-Net over F5 — Constructive and digital
Digital (84, 99, 55806)-net over F5, using
- 51 times duplication [i] based on digital (83, 98, 55806)-net over F5, using
- net defined by OOA [i] based on linear OOA(598, 55806, F5, 15, 15) (dual of [(55806, 15), 836992, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(598, 390643, F5, 15) (dual of [390643, 390545, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,6]) [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(581, 390626, F5, 13) (dual of [390626, 390545, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 390626 | 516−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (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([0,7]) ⊂ C([0,6]) [i] based on
- OOA 7-folding and stacking with additional row [i] based on linear OA(598, 390643, F5, 15) (dual of [390643, 390545, 16]-code), using
- net defined by OOA [i] based on linear OOA(598, 55806, F5, 15, 15) (dual of [(55806, 15), 836992, 16]-NRT-code), using
(84, 84+15, 263324)-Net over F5 — Digital
Digital (84, 99, 263324)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(599, 263324, F5, 15) (dual of [263324, 263225, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(599, 390645, F5, 15) (dual of [390645, 390546, 16]-code), using
- 1 times code embedding in larger space [i] based on linear OA(598, 390644, F5, 15) (dual of [390644, 390546, 16]-code), using
- construction X4 applied to C([0,7]) ⊂ C([0,6]) [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(581, 390626, F5, 13) (dual of [390626, 390545, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 390626 | 516−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(517, 18, F5, 17) (dual of [18, 1, 18]-code or 18-arc in PG(16,5)), using
- dual of repetition code with length 18 [i]
- linear OA(51, 18, F5, 1) (dual of [18, 17, 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
- 1 times code embedding in larger space [i] based on linear OA(598, 390644, F5, 15) (dual of [390644, 390546, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(599, 390645, F5, 15) (dual of [390645, 390546, 16]-code), using
(84, 84+15, large)-Net in Base 5 — Upper bound on s
There is no (84, 99, large)-net in base 5, because
- 13 times m-reduction [i] would yield (84, 86, large)-net in base 5, but