Best Known (29, 44, s)-Nets in Base 81
(29, 44, 75921)-Net over F81 — Constructive and digital
Digital (29, 44, 75921)-net over F81, using
- net defined by OOA [i] based on linear OOA(8144, 75921, F81, 15, 15) (dual of [(75921, 15), 1138771, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(8144, 531448, F81, 15) (dual of [531448, 531404, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(8144, 531449, F81, 15) (dual of [531449, 531405, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,6]) [i] based on
- 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(8137, 531442, F81, 13) (dual of [531442, 531405, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 816−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (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([0,7]) ⊂ C([0,6]) [i] based on
- discarding factors / shortening the dual code based on linear OA(8144, 531449, F81, 15) (dual of [531449, 531405, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(8144, 531448, F81, 15) (dual of [531448, 531404, 16]-code), using
(29, 44, 265724)-Net over F81 — Digital
Digital (29, 44, 265724)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8144, 265724, F81, 2, 15) (dual of [(265724, 2), 531404, 16]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8144, 531448, F81, 15) (dual of [531448, 531404, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(8144, 531449, F81, 15) (dual of [531449, 531405, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,6]) [i] based on
- 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(8137, 531442, F81, 13) (dual of [531442, 531405, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 816−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (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([0,7]) ⊂ C([0,6]) [i] based on
- discarding factors / shortening the dual code based on linear OA(8144, 531449, F81, 15) (dual of [531449, 531405, 16]-code), using
- OOA 2-folding [i] based on linear OA(8144, 531448, F81, 15) (dual of [531448, 531404, 16]-code), using
(29, 44, large)-Net in Base 81 — Upper bound on s
There is no (29, 44, large)-net in base 81, because
- 13 times m-reduction [i] would yield (29, 31, large)-net in base 81, but