Best Known (7, 7+7, s)-Nets in Base 81
(7, 7+7, 2188)-Net over F81 — Constructive and digital
Digital (7, 14, 2188)-net over F81, using
- net defined by OOA [i] based on linear OOA(8114, 2188, F81, 7, 7) (dual of [(2188, 7), 15302, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(8114, 6565, F81, 7) (dual of [6565, 6551, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(8114, 6567, F81, 7) (dual of [6567, 6553, 8]-code), using
- construction X applied to C([0,3]) ⊂ C([0,2]) [i] based on
- linear OA(8113, 6562, F81, 7) (dual of [6562, 6549, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- linear OA(819, 6562, F81, 5) (dual of [6562, 6553, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- linear OA(811, 5, F81, 1) (dual of [5, 4, 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([0,3]) ⊂ C([0,2]) [i] based on
- discarding factors / shortening the dual code based on linear OA(8114, 6567, F81, 7) (dual of [6567, 6553, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(8114, 6565, F81, 7) (dual of [6565, 6551, 8]-code), using
(7, 7+7, 3283)-Net over F81 — Digital
Digital (7, 14, 3283)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8114, 3283, F81, 2, 7) (dual of [(3283, 2), 6552, 8]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8114, 6566, F81, 7) (dual of [6566, 6552, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(8114, 6567, F81, 7) (dual of [6567, 6553, 8]-code), using
- construction X applied to C([0,3]) ⊂ C([0,2]) [i] based on
- linear OA(8113, 6562, F81, 7) (dual of [6562, 6549, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- linear OA(819, 6562, F81, 5) (dual of [6562, 6553, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- linear OA(811, 5, F81, 1) (dual of [5, 4, 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([0,3]) ⊂ C([0,2]) [i] based on
- discarding factors / shortening the dual code based on linear OA(8114, 6567, F81, 7) (dual of [6567, 6553, 8]-code), using
- OOA 2-folding [i] based on linear OA(8114, 6566, F81, 7) (dual of [6566, 6552, 8]-code), using
(7, 7+7, 4230536)-Net in Base 81 — Upper bound on s
There is no (7, 14, 4230537)-net in base 81, because
- 1 times m-reduction [i] would yield (7, 13, 4230537)-net in base 81, but
- the generalized Rao bound for nets shows that 81m ≥ 6 461086 022592 743283 994481 > 8113 [i]