Best Known (29, 42, s)-Nets in Base 81
(29, 42, 88577)-Net over F81 — Constructive and digital
Digital (29, 42, 88577)-net over F81, using
- net defined by OOA [i] based on linear OOA(8142, 88577, F81, 13, 13) (dual of [(88577, 13), 1151459, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(8142, 531463, F81, 13) (dual of [531463, 531421, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(8142, 531465, F81, 13) (dual of [531465, 531423, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([0,3]) [i] based on
- 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(8119, 531442, F81, 7) (dual of [531442, 531423, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 816−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- linear OA(815, 23, F81, 5) (dual of [23, 18, 6]-code or 23-arc in PG(4,81)), using
- discarding factors / shortening the dual code based on linear OA(815, 81, F81, 5) (dual of [81, 76, 6]-code or 81-arc in PG(4,81)), using
- Reed–Solomon code RS(76,81) [i]
- discarding factors / shortening the dual code based on linear OA(815, 81, F81, 5) (dual of [81, 76, 6]-code or 81-arc in PG(4,81)), using
- construction X applied to C([0,6]) ⊂ C([0,3]) [i] based on
- discarding factors / shortening the dual code based on linear OA(8142, 531465, F81, 13) (dual of [531465, 531423, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(8142, 531463, F81, 13) (dual of [531463, 531421, 14]-code), using
(29, 42, 531465)-Net over F81 — Digital
Digital (29, 42, 531465)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8142, 531465, F81, 13) (dual of [531465, 531423, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([0,3]) [i] based on
- 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(8119, 531442, F81, 7) (dual of [531442, 531423, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 816−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- linear OA(815, 23, F81, 5) (dual of [23, 18, 6]-code or 23-arc in PG(4,81)), using
- discarding factors / shortening the dual code based on linear OA(815, 81, F81, 5) (dual of [81, 76, 6]-code or 81-arc in PG(4,81)), using
- Reed–Solomon code RS(76,81) [i]
- discarding factors / shortening the dual code based on linear OA(815, 81, F81, 5) (dual of [81, 76, 6]-code or 81-arc in PG(4,81)), using
- construction X applied to C([0,6]) ⊂ C([0,3]) [i] based on
(29, 42, large)-Net in Base 81 — Upper bound on s
There is no (29, 42, large)-net in base 81, because
- 11 times m-reduction [i] would yield (29, 31, large)-net in base 81, but