Best Known (83, 100, s)-Nets in Base 9
(83, 100, 132862)-Net over F9 — Constructive and digital
Digital (83, 100, 132862)-net over F9, using
- net defined by OOA [i] based on linear OOA(9100, 132862, F9, 17, 17) (dual of [(132862, 17), 2258554, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(9100, 1062897, F9, 17) (dual of [1062897, 1062797, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(9100, 1062898, F9, 17) (dual of [1062898, 1062798, 18]-code), using
- trace code [i] based on linear OA(8150, 531449, F81, 17) (dual of [531449, 531399, 18]-code), using
- construction X applied to C([0,8]) ⊂ C([0,7]) [i] based on
- linear OA(8149, 531442, F81, 17) (dual of [531442, 531393, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 816−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(8143, 531442, F81, 15) (dual of [531442, 531399, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 816−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(811, 7, F81, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(811, s, F81, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,8]) ⊂ C([0,7]) [i] based on
- trace code [i] based on linear OA(8150, 531449, F81, 17) (dual of [531449, 531399, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(9100, 1062898, F9, 17) (dual of [1062898, 1062798, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(9100, 1062897, F9, 17) (dual of [1062897, 1062797, 18]-code), using
(83, 100, 1062898)-Net over F9 — Digital
Digital (83, 100, 1062898)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(9100, 1062898, F9, 17) (dual of [1062898, 1062798, 18]-code), using
- trace code [i] based on linear OA(8150, 531449, F81, 17) (dual of [531449, 531399, 18]-code), using
- construction X applied to C([0,8]) ⊂ C([0,7]) [i] based on
- linear OA(8149, 531442, F81, 17) (dual of [531442, 531393, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 816−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(8143, 531442, F81, 15) (dual of [531442, 531399, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 816−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(811, 7, F81, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(811, s, F81, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,8]) ⊂ C([0,7]) [i] based on
- trace code [i] based on linear OA(8150, 531449, F81, 17) (dual of [531449, 531399, 18]-code), using
(83, 100, large)-Net in Base 9 — Upper bound on s
There is no (83, 100, large)-net in base 9, because
- 15 times m-reduction [i] would yield (83, 85, large)-net in base 9, but