Best Known (69−23, 69, s)-Nets in Base 81
(69−23, 69, 48313)-Net over F81 — Constructive and digital
Digital (46, 69, 48313)-net over F81, using
- 812 times duplication [i] based on digital (44, 67, 48313)-net over F81, using
- net defined by OOA [i] based on linear OOA(8167, 48313, F81, 23, 23) (dual of [(48313, 23), 1111132, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(8167, 531444, F81, 23) (dual of [531444, 531377, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(21) [i] based on
- linear OA(8167, 531441, F81, 23) (dual of [531441, 531374, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(8164, 531441, F81, 22) (dual of [531441, 531377, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(810, 3, F81, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(810, s, F81, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(22) ⊂ Ce(21) [i] based on
- OOA 11-folding and stacking with additional row [i] based on linear OA(8167, 531444, F81, 23) (dual of [531444, 531377, 24]-code), using
- net defined by OOA [i] based on linear OOA(8167, 48313, F81, 23, 23) (dual of [(48313, 23), 1111132, 24]-NRT-code), using
(69−23, 69, 256818)-Net over F81 — Digital
Digital (46, 69, 256818)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8169, 256818, F81, 2, 23) (dual of [(256818, 2), 513567, 24]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8169, 265726, F81, 2, 23) (dual of [(265726, 2), 531383, 24]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8169, 531452, F81, 23) (dual of [531452, 531383, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(19) [i] based on
- linear OA(8167, 531441, F81, 23) (dual of [531441, 531374, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(8158, 531441, F81, 20) (dual of [531441, 531383, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [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(22) ⊂ Ce(19) [i] based on
- OOA 2-folding [i] based on linear OA(8169, 531452, F81, 23) (dual of [531452, 531383, 24]-code), using
- discarding factors / shortening the dual code based on linear OOA(8169, 265726, F81, 2, 23) (dual of [(265726, 2), 531383, 24]-NRT-code), using
(69−23, 69, large)-Net in Base 81 — Upper bound on s
There is no (46, 69, large)-net in base 81, because
- 21 times m-reduction [i] would yield (46, 48, large)-net in base 81, but