Best Known (44, 44+22, s)-Nets in Base 81
(44, 44+22, 48313)-Net over F81 — Constructive and digital
Digital (44, 66, 48313)-net over F81, using
- 1 times m-reduction [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
(44, 44+22, 265726)-Net over F81 — Digital
Digital (44, 66, 265726)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8166, 265726, F81, 2, 22) (dual of [(265726, 2), 531386, 23]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8166, 531452, F81, 22) (dual of [531452, 531386, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(18) [i] based on
- 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(8155, 531441, F81, 19) (dual of [531441, 531386, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [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(21) ⊂ Ce(18) [i] based on
- OOA 2-folding [i] based on linear OA(8166, 531452, F81, 22) (dual of [531452, 531386, 23]-code), using
(44, 44+22, large)-Net in Base 81 — Upper bound on s
There is no (44, 66, large)-net in base 81, because
- 20 times m-reduction [i] would yield (44, 46, large)-net in base 81, but