Best Known (20, 20+9, s)-Nets in Base 81
(20, 20+9, 132942)-Net over F81 — Constructive and digital
Digital (20, 29, 132942)-net over F81, using
- net defined by OOA [i] based on linear OOA(8129, 132942, F81, 12, 9) (dual of [(132942, 12), 1595275, 10]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(8129, 132943, F81, 4, 9) (dual of [(132943, 4), 531743, 10]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(814, 82, F81, 4, 4) (dual of [(82, 4), 324, 5]-NRT-code), using
- extended Reed–Solomon NRT-code RSe(4;324,81) [i]
- linear OOA(8125, 132861, F81, 4, 9) (dual of [(132861, 4), 531419, 10]-NRT-code), using
- OOA 4-folding [i] based on linear OA(8125, 531444, F81, 9) (dual of [531444, 531419, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(7) [i] based on
- 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(8122, 531441, F81, 8) (dual of [531441, 531419, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [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(8) ⊂ Ce(7) [i] based on
- OOA 4-folding [i] based on linear OA(8125, 531444, F81, 9) (dual of [531444, 531419, 10]-code), using
- linear OOA(814, 82, F81, 4, 4) (dual of [(82, 4), 324, 5]-NRT-code), using
- (u, u+v)-construction [i] based on
- OOA stacking with additional row [i] based on linear OOA(8129, 132943, F81, 4, 9) (dual of [(132943, 4), 531743, 10]-NRT-code), using
(20, 20+9, 531526)-Net over F81 — Digital
Digital (20, 29, 531526)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8129, 531526, F81, 9) (dual of [531526, 531497, 10]-code), using
- (u, u+v)-construction [i] based on
- linear OA(814, 82, F81, 4) (dual of [82, 78, 5]-code or 82-arc in PG(3,81)), using
- extended Reed–Solomon code RSe(78,81) [i]
- linear OA(8125, 531444, F81, 9) (dual of [531444, 531419, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(7) [i] based on
- 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(8122, 531441, F81, 8) (dual of [531441, 531419, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [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(8) ⊂ Ce(7) [i] based on
- linear OA(814, 82, F81, 4) (dual of [82, 78, 5]-code or 82-arc in PG(3,81)), using
- (u, u+v)-construction [i] based on
(20, 20+9, large)-Net in Base 81 — Upper bound on s
There is no (20, 29, large)-net in base 81, because
- 7 times m-reduction [i] would yield (20, 22, large)-net in base 81, but