Best Known (83, 98, s)-Nets in Base 5
(83, 98, 55806)-Net over F5 — Constructive and digital
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
(83, 98, 232660)-Net over F5 — Digital
Digital (83, 98, 232660)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(598, 232660, F5, 15) (dual of [232660, 232562, 16]-code), using
- discarding factors / shortening the dual code 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
- discarding factors / shortening the dual code based on linear OA(598, 390643, F5, 15) (dual of [390643, 390545, 16]-code), using
(83, 98, large)-Net in Base 5 — Upper bound on s
There is no (83, 98, large)-net in base 5, because
- 13 times m-reduction [i] would yield (83, 85, large)-net in base 5, but