Best Known (21, 42, s)-Nets in Base 81
(21, 42, 656)-Net over F81 — Constructive and digital
Digital (21, 42, 656)-net over F81, using
- 811 times duplication [i] based on digital (20, 41, 656)-net over F81, using
- net defined by OOA [i] based on linear OOA(8141, 656, F81, 21, 21) (dual of [(656, 21), 13735, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(8141, 6561, F81, 21) (dual of [6561, 6520, 22]-code), using
- an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- OOA 10-folding and stacking with additional row [i] based on linear OA(8141, 6561, F81, 21) (dual of [6561, 6520, 22]-code), using
- net defined by OOA [i] based on linear OOA(8141, 656, F81, 21, 21) (dual of [(656, 21), 13735, 22]-NRT-code), using
(21, 42, 2136)-Net over F81 — Digital
Digital (21, 42, 2136)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8142, 2136, F81, 3, 21) (dual of [(2136, 3), 6366, 22]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8142, 2189, F81, 3, 21) (dual of [(2189, 3), 6525, 22]-NRT-code), using
- OOA 3-folding [i] based on linear OA(8142, 6567, F81, 21) (dual of [6567, 6525, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,9]) [i] based on
- linear OA(8141, 6562, F81, 21) (dual of [6562, 6521, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(8137, 6562, F81, 19) (dual of [6562, 6525, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (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([0,10]) ⊂ C([0,9]) [i] based on
- OOA 3-folding [i] based on linear OA(8142, 6567, F81, 21) (dual of [6567, 6525, 22]-code), using
- discarding factors / shortening the dual code based on linear OOA(8142, 2189, F81, 3, 21) (dual of [(2189, 3), 6525, 22]-NRT-code), using
(21, 42, 3781589)-Net in Base 81 — Upper bound on s
There is no (21, 42, 3781590)-net in base 81, because
- 1 times m-reduction [i] would yield (21, 41, 3781590)-net in base 81, but
- the generalized Rao bound for nets shows that 81m ≥ 1 769646 645182 118575 231045 673091 289759 266481 394662 049972 142392 180662 817796 632001 > 8141 [i]