Best Known (71−23, 71, s)-Nets in Base 81
(71−23, 71, 48314)-Net over F81 — Constructive and digital
Digital (48, 71, 48314)-net over F81, using
- 811 times duplication [i] based on digital (47, 70, 48314)-net over F81, using
- net defined by OOA [i] based on linear OOA(8170, 48314, F81, 23, 23) (dual of [(48314, 23), 1111152, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(8170, 531455, F81, 23) (dual of [531455, 531385, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(8170, 531457, F81, 23) (dual of [531457, 531387, 24]-code), using
- construction X applied to C([0,11]) ⊂ C([0,9]) [i] based on
- linear OA(8167, 531442, F81, 23) (dual of [531442, 531375, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 816−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- linear OA(8155, 531442, F81, 19) (dual of [531442, 531387, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 816−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(813, 15, F81, 3) (dual of [15, 12, 4]-code or 15-arc in PG(2,81) or 15-cap in PG(2,81)), using
- discarding factors / shortening the dual code based on linear OA(813, 81, F81, 3) (dual of [81, 78, 4]-code or 81-arc in PG(2,81) or 81-cap in PG(2,81)), using
- Reed–Solomon code RS(78,81) [i]
- discarding factors / shortening the dual code based on linear OA(813, 81, F81, 3) (dual of [81, 78, 4]-code or 81-arc in PG(2,81) or 81-cap in PG(2,81)), using
- construction X applied to C([0,11]) ⊂ C([0,9]) [i] based on
- discarding factors / shortening the dual code based on linear OA(8170, 531457, F81, 23) (dual of [531457, 531387, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(8170, 531455, F81, 23) (dual of [531455, 531385, 24]-code), using
- net defined by OOA [i] based on linear OOA(8170, 48314, F81, 23, 23) (dual of [(48314, 23), 1111152, 24]-NRT-code), using
(71−23, 71, 265730)-Net over F81 — Digital
Digital (48, 71, 265730)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8171, 265730, F81, 2, 23) (dual of [(265730, 2), 531389, 24]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8171, 531460, F81, 23) (dual of [531460, 531389, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(17) [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(8152, 531441, F81, 18) (dual of [531441, 531389, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(814, 19, F81, 4) (dual of [19, 15, 5]-code or 19-arc in PG(3,81)), using
- discarding factors / shortening the dual code based on linear OA(814, 81, F81, 4) (dual of [81, 77, 5]-code or 81-arc in PG(3,81)), using
- Reed–Solomon code RS(77,81) [i]
- discarding factors / shortening the dual code based on linear OA(814, 81, F81, 4) (dual of [81, 77, 5]-code or 81-arc in PG(3,81)), using
- construction X applied to Ce(22) ⊂ Ce(17) [i] based on
- OOA 2-folding [i] based on linear OA(8171, 531460, F81, 23) (dual of [531460, 531389, 24]-code), using
(71−23, 71, large)-Net in Base 81 — Upper bound on s
There is no (48, 71, large)-net in base 81, because
- 21 times m-reduction [i] would yield (48, 50, large)-net in base 81, but