Best Known (27, 51, s)-Nets in Base 81
(27, 51, 547)-Net over F81 — Constructive and digital
Digital (27, 51, 547)-net over F81, using
- 811 times duplication [i] based on digital (26, 50, 547)-net over F81, using
- t-expansion [i] based on digital (25, 50, 547)-net over F81, using
- net defined by OOA [i] based on linear OOA(8150, 547, F81, 25, 25) (dual of [(547, 25), 13625, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(8150, 6565, F81, 25) (dual of [6565, 6515, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(8150, 6567, F81, 25) (dual of [6567, 6517, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,11]) [i] based on
- linear OA(8149, 6562, F81, 25) (dual of [6562, 6513, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(8145, 6562, F81, 23) (dual of [6562, 6517, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (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([0,12]) ⊂ C([0,11]) [i] based on
- discarding factors / shortening the dual code based on linear OA(8150, 6567, F81, 25) (dual of [6567, 6517, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(8150, 6565, F81, 25) (dual of [6565, 6515, 26]-code), using
- net defined by OOA [i] based on linear OOA(8150, 547, F81, 25, 25) (dual of [(547, 25), 13625, 26]-NRT-code), using
- t-expansion [i] based on digital (25, 50, 547)-net over F81, using
(27, 51, 3070)-Net over F81 — Digital
Digital (27, 51, 3070)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8151, 3070, F81, 2, 24) (dual of [(3070, 2), 6089, 25]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8151, 3287, F81, 2, 24) (dual of [(3287, 2), 6523, 25]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8151, 6574, F81, 24) (dual of [6574, 6523, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(8151, 6575, F81, 24) (dual of [6575, 6524, 25]-code), using
- construction X applied to Ce(23) ⊂ Ce(18) [i] based on
- linear OA(8147, 6561, F81, 24) (dual of [6561, 6514, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(8137, 6561, F81, 19) (dual of [6561, 6524, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(814, 14, F81, 4) (dual of [14, 10, 5]-code or 14-arc in PG(3,81)), using
- discarding factors / shortening the dual code based on linear OA(814, 81, F81, 4) (dual of [81, 77, 5]-code or 81-arc in PG(3,81)), using
- Reed–Solomon code RS(77,81) [i]
- discarding factors / shortening the dual code based on linear OA(814, 81, F81, 4) (dual of [81, 77, 5]-code or 81-arc in PG(3,81)), using
- construction X applied to Ce(23) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(8151, 6575, F81, 24) (dual of [6575, 6524, 25]-code), using
- OOA 2-folding [i] based on linear OA(8151, 6574, F81, 24) (dual of [6574, 6523, 25]-code), using
- discarding factors / shortening the dual code based on linear OOA(8151, 3287, F81, 2, 24) (dual of [(3287, 2), 6523, 25]-NRT-code), using
(27, 51, large)-Net in Base 81 — Upper bound on s
There is no (27, 51, large)-net in base 81, because
- 22 times m-reduction [i] would yield (27, 29, large)-net in base 81, but