Best Known (52−25, 52, s)-Nets in Base 81
(52−25, 52, 547)-Net over F81 — Constructive and digital
Digital (27, 52, 547)-net over F81, using
- 812 times duplication [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
(52−25, 52, 2451)-Net over F81 — Digital
Digital (27, 52, 2451)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8152, 2451, F81, 2, 25) (dual of [(2451, 2), 4850, 26]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8152, 3286, F81, 2, 25) (dual of [(3286, 2), 6520, 26]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8152, 6572, F81, 25) (dual of [6572, 6520, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(8152, 6573, F81, 25) (dual of [6573, 6521, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,10]) [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(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(813, 11, F81, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,81) or 11-cap in PG(2,81)), using
- discarding factors / shortening the dual code based on linear OA(813, 81, F81, 3) (dual of [81, 78, 4]-code or 81-arc in PG(2,81) or 81-cap in PG(2,81)), using
- Reed–Solomon code RS(78,81) [i]
- discarding factors / shortening the dual code based on linear OA(813, 81, F81, 3) (dual of [81, 78, 4]-code or 81-arc in PG(2,81) or 81-cap in PG(2,81)), using
- construction X applied to C([0,12]) ⊂ C([0,10]) [i] based on
- discarding factors / shortening the dual code based on linear OA(8152, 6573, F81, 25) (dual of [6573, 6521, 26]-code), using
- OOA 2-folding [i] based on linear OA(8152, 6572, F81, 25) (dual of [6572, 6520, 26]-code), using
- discarding factors / shortening the dual code based on linear OOA(8152, 3286, F81, 2, 25) (dual of [(3286, 2), 6520, 26]-NRT-code), using
(52−25, 52, large)-Net in Base 81 — Upper bound on s
There is no (27, 52, large)-net in base 81, because
- 23 times m-reduction [i] would yield (27, 29, large)-net in base 81, but