Best Known (41, 41+21, s)-Nets in Base 81
(41, 41+21, 53144)-Net over F81 — Constructive and digital
Digital (41, 62, 53144)-net over F81, using
- 811 times duplication [i] based on digital (40, 61, 53144)-net over F81, using
- net defined by OOA [i] based on linear OOA(8161, 53144, F81, 21, 21) (dual of [(53144, 21), 1115963, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(8161, 531441, F81, 21) (dual of [531441, 531380, 22]-code), using
- an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- OOA 10-folding and stacking with additional row [i] based on linear OA(8161, 531441, F81, 21) (dual of [531441, 531380, 22]-code), using
- net defined by OOA [i] based on linear OOA(8161, 53144, F81, 21, 21) (dual of [(53144, 21), 1115963, 22]-NRT-code), using
(41, 41+21, 217090)-Net over F81 — Digital
Digital (41, 62, 217090)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8162, 217090, F81, 2, 21) (dual of [(217090, 2), 434118, 22]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8162, 265724, F81, 2, 21) (dual of [(265724, 2), 531386, 22]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8162, 531448, F81, 21) (dual of [531448, 531386, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(8162, 531449, F81, 21) (dual of [531449, 531387, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,9]) [i] based on
- linear OA(8161, 531442, F81, 21) (dual of [531442, 531381, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 816−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (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(811, 7, F81, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(811, s, F81, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,10]) ⊂ C([0,9]) [i] based on
- discarding factors / shortening the dual code based on linear OA(8162, 531449, F81, 21) (dual of [531449, 531387, 22]-code), using
- OOA 2-folding [i] based on linear OA(8162, 531448, F81, 21) (dual of [531448, 531386, 22]-code), using
- discarding factors / shortening the dual code based on linear OOA(8162, 265724, F81, 2, 21) (dual of [(265724, 2), 531386, 22]-NRT-code), using
(41, 41+21, large)-Net in Base 81 — Upper bound on s
There is no (41, 62, large)-net in base 81, because
- 19 times m-reduction [i] would yield (41, 43, large)-net in base 81, but