Best Known (12, 23, s)-Nets in Base 81
(12, 23, 1313)-Net over F81 — Constructive and digital
Digital (12, 23, 1313)-net over F81, using
- 811 times duplication [i] based on digital (11, 22, 1313)-net over F81, using
- net defined by OOA [i] based on linear OOA(8122, 1313, F81, 11, 11) (dual of [(1313, 11), 14421, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(8122, 6566, F81, 11) (dual of [6566, 6544, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(8122, 6567, F81, 11) (dual of [6567, 6545, 12]-code), using
- construction X applied to C([0,5]) ⊂ C([0,4]) [i] based on
- linear OA(8121, 6562, F81, 11) (dual of [6562, 6541, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- 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(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([0,5]) ⊂ C([0,4]) [i] based on
- discarding factors / shortening the dual code based on linear OA(8122, 6567, F81, 11) (dual of [6567, 6545, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(8122, 6566, F81, 11) (dual of [6566, 6544, 12]-code), using
- net defined by OOA [i] based on linear OOA(8122, 1313, F81, 11, 11) (dual of [(1313, 11), 14421, 12]-NRT-code), using
(12, 23, 3284)-Net over F81 — Digital
Digital (12, 23, 3284)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8123, 3284, F81, 2, 11) (dual of [(3284, 2), 6545, 12]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8123, 6568, F81, 11) (dual of [6568, 6545, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(8123, 6569, F81, 11) (dual of [6569, 6546, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(7) [i] based on
- linear OA(8121, 6561, F81, 11) (dual of [6561, 6540, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(8115, 6561, F81, 8) (dual of [6561, 6546, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(812, 8, F81, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,81)), using
- discarding factors / shortening the dual code based on linear OA(812, 81, F81, 2) (dual of [81, 79, 3]-code or 81-arc in PG(1,81)), using
- Reed–Solomon code RS(79,81) [i]
- discarding factors / shortening the dual code based on linear OA(812, 81, F81, 2) (dual of [81, 79, 3]-code or 81-arc in PG(1,81)), using
- construction X applied to Ce(10) ⊂ Ce(7) [i] based on
- discarding factors / shortening the dual code based on linear OA(8123, 6569, F81, 11) (dual of [6569, 6546, 12]-code), using
- OOA 2-folding [i] based on linear OA(8123, 6568, F81, 11) (dual of [6568, 6545, 12]-code), using
(12, 23, 8129806)-Net in Base 81 — Upper bound on s
There is no (12, 23, 8129807)-net in base 81, because
- 1 times m-reduction [i] would yield (12, 22, 8129807)-net in base 81, but
- the generalized Rao bound for nets shows that 81m ≥ 969773 735895 027015 798059 765975 863901 394801 > 8122 [i]