Best Known (53−23, 53, s)-Nets in Base 81
(53−23, 53, 598)-Net over F81 — Constructive and digital
Digital (30, 53, 598)-net over F81, using
- 813 times duplication [i] based on digital (27, 50, 598)-net over F81, using
- net defined by OOA [i] based on linear OOA(8150, 598, F81, 23, 23) (dual of [(598, 23), 13704, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(8150, 6579, F81, 23) (dual of [6579, 6529, 24]-code), using
- construction X applied to C([0,11]) ⊂ C([0,8]) [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(8133, 6562, F81, 17) (dual of [6562, 6529, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(815, 17, F81, 5) (dual of [17, 12, 6]-code or 17-arc in PG(4,81)), using
- discarding factors / shortening the dual code based on linear OA(815, 81, F81, 5) (dual of [81, 76, 6]-code or 81-arc in PG(4,81)), using
- Reed–Solomon code RS(76,81) [i]
- discarding factors / shortening the dual code based on linear OA(815, 81, F81, 5) (dual of [81, 76, 6]-code or 81-arc in PG(4,81)), using
- construction X applied to C([0,11]) ⊂ C([0,8]) [i] based on
- OOA 11-folding and stacking with additional row [i] based on linear OA(8150, 6579, F81, 23) (dual of [6579, 6529, 24]-code), using
- net defined by OOA [i] based on linear OOA(8150, 598, F81, 23, 23) (dual of [(598, 23), 13704, 24]-NRT-code), using
(53−23, 53, 5760)-Net over F81 — Digital
Digital (30, 53, 5760)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8153, 5760, F81, 23) (dual of [5760, 5707, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(8153, 6587, F81, 23) (dual of [6587, 6534, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(13) [i] based on
- linear OA(8145, 6561, F81, 23) (dual of [6561, 6516, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(8127, 6561, F81, 14) (dual of [6561, 6534, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(818, 26, F81, 8) (dual of [26, 18, 9]-code or 26-arc in PG(7,81)), using
- discarding factors / shortening the dual code based on linear OA(818, 81, F81, 8) (dual of [81, 73, 9]-code or 81-arc in PG(7,81)), using
- Reed–Solomon code RS(73,81) [i]
- discarding factors / shortening the dual code based on linear OA(818, 81, F81, 8) (dual of [81, 73, 9]-code or 81-arc in PG(7,81)), using
- construction X applied to Ce(22) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(8153, 6587, F81, 23) (dual of [6587, 6534, 24]-code), using
(53−23, 53, large)-Net in Base 81 — Upper bound on s
There is no (30, 53, large)-net in base 81, because
- 21 times m-reduction [i] would yield (30, 32, large)-net in base 81, but