Best Known (9, 18, s)-Nets in Base 81
(9, 18, 1641)-Net over F81 — Constructive and digital
Digital (9, 18, 1641)-net over F81, using
- net defined by OOA [i] based on linear OOA(8118, 1641, F81, 9, 9) (dual of [(1641, 9), 14751, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(8118, 6565, F81, 9) (dual of [6565, 6547, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(8118, 6567, F81, 9) (dual of [6567, 6549, 10]-code), using
- construction X applied to C([0,4]) ⊂ C([0,3]) [i] based on
- linear OA(8117, 6562, F81, 9) (dual of [6562, 6545, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- 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(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([0,4]) ⊂ C([0,3]) [i] based on
- discarding factors / shortening the dual code based on linear OA(8118, 6567, F81, 9) (dual of [6567, 6549, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(8118, 6565, F81, 9) (dual of [6565, 6547, 10]-code), using
(9, 18, 3283)-Net over F81 — Digital
Digital (9, 18, 3283)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8118, 3283, F81, 2, 9) (dual of [(3283, 2), 6548, 10]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8118, 6566, F81, 9) (dual of [6566, 6548, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(8118, 6567, F81, 9) (dual of [6567, 6549, 10]-code), using
- construction X applied to C([0,4]) ⊂ C([0,3]) [i] based on
- linear OA(8117, 6562, F81, 9) (dual of [6562, 6545, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- 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(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([0,4]) ⊂ C([0,3]) [i] based on
- discarding factors / shortening the dual code based on linear OA(8118, 6567, F81, 9) (dual of [6567, 6549, 10]-code), using
- OOA 2-folding [i] based on linear OA(8118, 6566, F81, 9) (dual of [6566, 6548, 10]-code), using
(9, 18, 3572925)-Net in Base 81 — Upper bound on s
There is no (9, 18, 3572926)-net in base 81, because
- 1 times m-reduction [i] would yield (9, 17, 3572926)-net in base 81, but
- the generalized Rao bound for nets shows that 81m ≥ 278 128533 459560 965457 708317 262721 > 8117 [i]