Best Known (31, 31+23, s)-Nets in Base 81
(31, 31+23, 599)-Net over F81 — Constructive and digital
Digital (31, 54, 599)-net over F81, using
- net defined by OOA [i] based on linear OOA(8154, 599, F81, 23, 23) (dual of [(599, 23), 13723, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(8154, 6590, F81, 23) (dual of [6590, 6536, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(8154, 6591, F81, 23) (dual of [6591, 6537, 24]-code), using
- construction X applied to C([0,11]) ⊂ C([0,6]) [i] based on
- 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(8125, 6562, F81, 13) (dual of [6562, 6537, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(819, 29, F81, 9) (dual of [29, 20, 10]-code or 29-arc in PG(8,81)), using
- discarding factors / shortening the dual code based on linear OA(819, 81, F81, 9) (dual of [81, 72, 10]-code or 81-arc in PG(8,81)), using
- Reed–Solomon code RS(72,81) [i]
- discarding factors / shortening the dual code based on linear OA(819, 81, F81, 9) (dual of [81, 72, 10]-code or 81-arc in PG(8,81)), using
- construction X applied to C([0,11]) ⊂ C([0,6]) [i] based on
- discarding factors / shortening the dual code based on linear OA(8154, 6591, F81, 23) (dual of [6591, 6537, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(8154, 6590, F81, 23) (dual of [6590, 6536, 24]-code), using
(31, 31+23, 6591)-Net over F81 — Digital
Digital (31, 54, 6591)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8154, 6591, F81, 23) (dual of [6591, 6537, 24]-code), using
- construction X applied to C([0,11]) ⊂ C([0,6]) [i] based on
- 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(8125, 6562, F81, 13) (dual of [6562, 6537, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(819, 29, F81, 9) (dual of [29, 20, 10]-code or 29-arc in PG(8,81)), using
- discarding factors / shortening the dual code based on linear OA(819, 81, F81, 9) (dual of [81, 72, 10]-code or 81-arc in PG(8,81)), using
- Reed–Solomon code RS(72,81) [i]
- discarding factors / shortening the dual code based on linear OA(819, 81, F81, 9) (dual of [81, 72, 10]-code or 81-arc in PG(8,81)), using
- construction X applied to C([0,11]) ⊂ C([0,6]) [i] based on
(31, 31+23, large)-Net in Base 81 — Upper bound on s
There is no (31, 54, large)-net in base 81, because
- 21 times m-reduction [i] would yield (31, 33, large)-net in base 81, but