Best Known (84−13, 84, s)-Nets in Base 5
(84−13, 84, 65107)-Net over F5 — Constructive and digital
Digital (71, 84, 65107)-net over F5, using
- 52 times duplication [i] based on digital (69, 82, 65107)-net over F5, using
- net defined by OOA [i] based on linear OOA(582, 65107, F5, 13, 13) (dual of [(65107, 13), 846309, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(582, 390643, F5, 13) (dual of [390643, 390561, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([0,5]) [i] based on
- 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(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(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([0,6]) ⊂ C([0,5]) [i] based on
- OOA 6-folding and stacking with additional row [i] based on linear OA(582, 390643, F5, 13) (dual of [390643, 390561, 14]-code), using
- net defined by OOA [i] based on linear OOA(582, 65107, F5, 13, 13) (dual of [(65107, 13), 846309, 14]-NRT-code), using
(84−13, 84, 230668)-Net over F5 — Digital
Digital (71, 84, 230668)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(584, 230668, F5, 13) (dual of [230668, 230584, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(584, 390646, F5, 13) (dual of [390646, 390562, 14]-code), using
- 2 times code embedding in larger space [i] based on linear OA(582, 390644, F5, 13) (dual of [390644, 390562, 14]-code), using
- construction X4 applied to C([0,6]) ⊂ C([0,5]) [i] based on
- 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(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(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([0,6]) ⊂ C([0,5]) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(582, 390644, F5, 13) (dual of [390644, 390562, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(584, 390646, F5, 13) (dual of [390646, 390562, 14]-code), using
(84−13, 84, large)-Net in Base 5 — Upper bound on s
There is no (71, 84, large)-net in base 5, because
- 11 times m-reduction [i] would yield (71, 73, large)-net in base 5, but