Best Known (58−28, 58, s)-Nets in Base 81
(58−28, 58, 469)-Net over F81 — Constructive and digital
Digital (30, 58, 469)-net over F81, using
- t-expansion [i] based on digital (29, 58, 469)-net over F81, using
- net defined by OOA [i] based on linear OOA(8158, 469, F81, 29, 29) (dual of [(469, 29), 13543, 30]-NRT-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(8158, 6567, F81, 29) (dual of [6567, 6509, 30]-code), using
- construction X applied to C([0,14]) ⊂ C([0,13]) [i] based on
- linear OA(8157, 6562, F81, 29) (dual of [6562, 6505, 30]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,14], and minimum distance d ≥ |{−14,−13,…,14}|+1 = 30 (BCH-bound) [i]
- linear OA(8153, 6562, F81, 27) (dual of [6562, 6509, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (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([0,14]) ⊂ C([0,13]) [i] based on
- OOA 14-folding and stacking with additional row [i] based on linear OA(8158, 6567, F81, 29) (dual of [6567, 6509, 30]-code), using
- net defined by OOA [i] based on linear OOA(8158, 469, F81, 29, 29) (dual of [(469, 29), 13543, 30]-NRT-code), using
(58−28, 58, 2385)-Net over F81 — Digital
Digital (30, 58, 2385)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8158, 2385, F81, 2, 28) (dual of [(2385, 2), 4712, 29]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8158, 3286, F81, 2, 28) (dual of [(3286, 2), 6514, 29]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8158, 6572, F81, 28) (dual of [6572, 6514, 29]-code), using
- construction X applied to Ce(27) ⊂ Ce(23) [i] based on
- linear OA(8155, 6561, F81, 28) (dual of [6561, 6506, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- 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(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 Ce(27) ⊂ Ce(23) [i] based on
- OOA 2-folding [i] based on linear OA(8158, 6572, F81, 28) (dual of [6572, 6514, 29]-code), using
- discarding factors / shortening the dual code based on linear OOA(8158, 3286, F81, 2, 28) (dual of [(3286, 2), 6514, 29]-NRT-code), using
(58−28, 58, 6094640)-Net in Base 81 — Upper bound on s
There is no (30, 58, 6094641)-net in base 81, because
- the generalized Rao bound for nets shows that 81m ≥ 492 189030 192746 261768 076643 051619 055814 248052 872992 354854 871615 258686 139612 481614 790997 456818 752701 065255 252321 > 8158 [i]