Best Known (33−10, 33, s)-Nets in Base 81
(33−10, 33, 106370)-Net over F81 — Constructive and digital
Digital (23, 33, 106370)-net over F81, using
- (u, u+v)-construction [i] based on
- digital (0, 5, 82)-net over F81, using
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 0 and N(F) ≥ 82, using
- the rational function field F81(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- digital (18, 28, 106288)-net over F81, using
- net defined by OOA [i] based on linear OOA(8128, 106288, F81, 10, 10) (dual of [(106288, 10), 1062852, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(8128, 531440, F81, 10) (dual of [531440, 531412, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(8128, 531441, F81, 10) (dual of [531441, 531413, 11]-code), using
- an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- discarding factors / shortening the dual code based on linear OA(8128, 531441, F81, 10) (dual of [531441, 531413, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(8128, 531440, F81, 10) (dual of [531440, 531412, 11]-code), using
- net defined by OOA [i] based on linear OOA(8128, 106288, F81, 10, 10) (dual of [(106288, 10), 1062852, 11]-NRT-code), using
- digital (0, 5, 82)-net over F81, using
(33−10, 33, 531526)-Net over F81 — Digital
Digital (23, 33, 531526)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8133, 531526, F81, 10) (dual of [531526, 531493, 11]-code), using
- (u, u+v)-construction [i] based on
- linear OA(815, 82, F81, 5) (dual of [82, 77, 6]-code or 82-arc in PG(4,81)), using
- extended Reed–Solomon code RSe(77,81) [i]
- the expurgated narrow-sense BCH-code C(I) with length 82 | 812−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- linear OA(8128, 531444, F81, 10) (dual of [531444, 531416, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(8) [i] based on
- linear OA(8128, 531441, F81, 10) (dual of [531441, 531413, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [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(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(9) ⊂ Ce(8) [i] based on
- linear OA(815, 82, F81, 5) (dual of [82, 77, 6]-code or 82-arc in PG(4,81)), using
- (u, u+v)-construction [i] based on
(33−10, 33, large)-Net in Base 81 — Upper bound on s
There is no (23, 33, large)-net in base 81, because
- 8 times m-reduction [i] would yield (23, 25, large)-net in base 81, but