Best Known (33, 33+17, s)-Nets in Base 81
(33, 33+17, 66431)-Net over F81 — Constructive and digital
Digital (33, 50, 66431)-net over F81, using
- net defined by OOA [i] based on linear OOA(8150, 66431, F81, 17, 17) (dual of [(66431, 17), 1129277, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [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
- OOA 8-folding and stacking with additional row [i] based on linear OA(8150, 531449, F81, 17) (dual of [531449, 531399, 18]-code), using
(33, 33+17, 264080)-Net over F81 — Digital
Digital (33, 50, 264080)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8150, 264080, F81, 2, 17) (dual of [(264080, 2), 528110, 18]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8150, 265724, F81, 2, 17) (dual of [(265724, 2), 531398, 18]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8150, 531448, F81, 17) (dual of [531448, 531398, 18]-code), using
- discarding factors / shortening the dual code 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
- discarding factors / shortening the dual code based on linear OA(8150, 531449, F81, 17) (dual of [531449, 531399, 18]-code), using
- OOA 2-folding [i] based on linear OA(8150, 531448, F81, 17) (dual of [531448, 531398, 18]-code), using
- discarding factors / shortening the dual code based on linear OOA(8150, 265724, F81, 2, 17) (dual of [(265724, 2), 531398, 18]-NRT-code), using
(33, 33+17, large)-Net in Base 81 — Upper bound on s
There is no (33, 50, large)-net in base 81, because
- 15 times m-reduction [i] would yield (33, 35, large)-net in base 81, but