Best Known (11, 11+11, s)-Nets in Base 81
(11, 11+11, 1313)-Net over F81 — Constructive and digital
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
(11, 11+11, 2775)-Net over F81 — Digital
Digital (11, 22, 2775)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8122, 2775, F81, 2, 11) (dual of [(2775, 2), 5528, 12]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8122, 3283, F81, 2, 11) (dual of [(3283, 2), 6544, 12]-NRT-code), using
- OOA 2-folding [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 2-folding [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 OOA(8122, 3283, F81, 2, 11) (dual of [(3283, 2), 6544, 12]-NRT-code), using
(11, 11+11, 3375849)-Net in Base 81 — Upper bound on s
There is no (11, 22, 3375850)-net in base 81, because
- 1 times m-reduction [i] would yield (11, 21, 3375850)-net in base 81, but
- the generalized Rao bound for nets shows that 81m ≥ 11972 524154 855054 234698 718312 509457 540001 > 8121 [i]