Best Known (57, 57+5, s)-Nets in Base 81
(57, 57+5, large)-Net over F81 — Constructive and digital
Digital (57, 62, large)-net over F81, using
- t-expansion [i] based on digital (53, 62, large)-net over F81, using
- (u, u+v)-construction [i] based on
- digital (9, 13, 4194301)-net over F81, using
- net defined by OOA [i] based on linear OOA(8113, 4194301, F81, 4, 4) (dual of [(4194301, 4), 16777191, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(8113, 8388602, F81, 4) (dual of [8388602, 8388589, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(8113, large, F81, 4) (dual of [large, large−13, 5]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523360 | 814−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 5 [i]
- discarding factors / shortening the dual code based on linear OA(8113, large, F81, 4) (dual of [large, large−13, 5]-code), using
- OA 2-folding and stacking [i] based on linear OA(8113, 8388602, F81, 4) (dual of [8388602, 8388589, 5]-code), using
- net defined by OOA [i] based on linear OOA(8113, 4194301, F81, 4, 4) (dual of [(4194301, 4), 16777191, 5]-NRT-code), using
- digital (40, 49, 4194381)-net over F81, using
- net defined by OOA [i] based on linear OOA(8149, 4194381, F81, 12, 9) (dual of [(4194381, 12), 50332523, 10]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(8149, 4194382, F81, 4, 9) (dual of [(4194382, 4), 16777479, 10]-NRT-code), using
- generalized (u, u+v)-construction [i] based on
- linear OOA(813, 82, F81, 4, 3) (dual of [(82, 4), 325, 4]-NRT-code), using
- extended Reed–Solomon NRT-code RSe(4;325,81) [i]
- linear OOA(8113, 2097150, F81, 4, 4) (dual of [(2097150, 4), 8388587, 5]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8113, 4194301, F81, 4, 4) (dual of [(4194301, 4), 16777191, 5]-NRT-code) (see above)
- linear OOA(8133, 2097150, F81, 4, 9) (dual of [(2097150, 4), 8388567, 10]-NRT-code), using
- OOA 4-folding [i] based on linear OA(8133, 8388600, F81, 9) (dual of [8388600, 8388567, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(8133, large, F81, 9) (dual of [large, large−33, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523361 | 818−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(8133, large, F81, 9) (dual of [large, large−33, 10]-code), using
- OOA 4-folding [i] based on linear OA(8133, 8388600, F81, 9) (dual of [8388600, 8388567, 10]-code), using
- linear OOA(813, 82, F81, 4, 3) (dual of [(82, 4), 325, 4]-NRT-code), using
- generalized (u, u+v)-construction [i] based on
- OOA stacking with additional row [i] based on linear OOA(8149, 4194382, F81, 4, 9) (dual of [(4194382, 4), 16777479, 10]-NRT-code), using
- net defined by OOA [i] based on linear OOA(8149, 4194381, F81, 12, 9) (dual of [(4194381, 12), 50332523, 10]-NRT-code), using
- digital (9, 13, 4194301)-net over F81, using
- (u, u+v)-construction [i] based on
(57, 57+5, large)-Net in Base 81 — Upper bound on s
There is no (57, 62, large)-net in base 81, because
- 3 times m-reduction [i] would yield (57, 59, large)-net in base 81, but