Best Known (36−12, 36, s)-Nets in Base 81
(36−12, 36, 88575)-Net over F81 — Constructive and digital
Digital (24, 36, 88575)-net over F81, using
- net defined by OOA [i] based on linear OOA(8136, 88575, F81, 12, 12) (dual of [(88575, 12), 1062864, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(8136, 531450, F81, 12) (dual of [531450, 531414, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(8136, 531452, F81, 12) (dual of [531452, 531416, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(8) [i] based on
- linear OA(8134, 531441, F81, 12) (dual of [531441, 531407, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(8125, 531441, F81, 9) (dual of [531441, 531416, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(812, 11, F81, 2) (dual of [11, 9, 3]-code or 11-arc in PG(1,81)), using
- discarding factors / shortening the dual code based on linear OA(812, 81, F81, 2) (dual of [81, 79, 3]-code or 81-arc in PG(1,81)), using
- Reed–Solomon code RS(79,81) [i]
- discarding factors / shortening the dual code based on linear OA(812, 81, F81, 2) (dual of [81, 79, 3]-code or 81-arc in PG(1,81)), using
- construction X applied to Ce(11) ⊂ Ce(8) [i] based on
- discarding factors / shortening the dual code based on linear OA(8136, 531452, F81, 12) (dual of [531452, 531416, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(8136, 531450, F81, 12) (dual of [531450, 531414, 13]-code), using
(36−12, 36, 270755)-Net over F81 — Digital
Digital (24, 36, 270755)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8136, 270755, F81, 12) (dual of [270755, 270719, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(8136, 531452, F81, 12) (dual of [531452, 531416, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(8) [i] based on
- linear OA(8134, 531441, F81, 12) (dual of [531441, 531407, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(8125, 531441, F81, 9) (dual of [531441, 531416, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(812, 11, F81, 2) (dual of [11, 9, 3]-code or 11-arc in PG(1,81)), using
- discarding factors / shortening the dual code based on linear OA(812, 81, F81, 2) (dual of [81, 79, 3]-code or 81-arc in PG(1,81)), using
- Reed–Solomon code RS(79,81) [i]
- discarding factors / shortening the dual code based on linear OA(812, 81, F81, 2) (dual of [81, 79, 3]-code or 81-arc in PG(1,81)), using
- construction X applied to Ce(11) ⊂ Ce(8) [i] based on
- discarding factors / shortening the dual code based on linear OA(8136, 531452, F81, 12) (dual of [531452, 531416, 13]-code), using
(36−12, 36, large)-Net in Base 81 — Upper bound on s
There is no (24, 36, large)-net in base 81, because
- 10 times m-reduction [i] would yield (24, 26, large)-net in base 81, but