Best Known (32, 32+8, s)-Nets in Base 81
(32, 32+8, 2362874)-Net over F81 — Constructive and digital
Digital (32, 40, 2362874)-net over F81, using
- (u, u+v)-construction [i] based on
- digital (7, 11, 265724)-net over F81, using
- net defined by OOA [i] based on linear OOA(8111, 265724, F81, 4, 4) (dual of [(265724, 4), 1062885, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(8111, 531448, F81, 4) (dual of [531448, 531437, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(1) [i] based on
- linear OA(8110, 531441, F81, 4) (dual of [531441, 531431, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(814, 531441, F81, 2) (dual of [531441, 531437, 3]-code), using an extension Ce(1) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,1], and designed minimum distance d ≥ |I|+1 = 2 [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 Ce(3) ⊂ Ce(1) [i] based on
- OA 2-folding and stacking [i] based on linear OA(8111, 531448, F81, 4) (dual of [531448, 531437, 5]-code), using
- net defined by OOA [i] based on linear OOA(8111, 265724, F81, 4, 4) (dual of [(265724, 4), 1062885, 5]-NRT-code), using
- digital (21, 29, 2097150)-net over F81, using
- net defined by OOA [i] based on linear OOA(8129, 2097150, F81, 8, 8) (dual of [(2097150, 8), 16777171, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(8129, 8388600, F81, 8) (dual of [8388600, 8388571, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(8129, large, F81, 8) (dual of [large, large−29, 9]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523360 | 814−1, defining interval I = [0,7], and designed minimum distance d ≥ |I|+1 = 9 [i]
- discarding factors / shortening the dual code based on linear OA(8129, large, F81, 8) (dual of [large, large−29, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(8129, 8388600, F81, 8) (dual of [8388600, 8388571, 9]-code), using
- net defined by OOA [i] based on linear OOA(8129, 2097150, F81, 8, 8) (dual of [(2097150, 8), 16777171, 9]-NRT-code), using
- digital (7, 11, 265724)-net over F81, using
(32, 32+8, large)-Net over F81 — Digital
Digital (32, 40, large)-net over F81, using
- t-expansion [i] based on digital (30, 40, large)-net over F81, using
- 1 times m-reduction [i] based on digital (30, 41, large)-net over F81, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8141, large, F81, 11) (dual of [large, large−41, 12]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523361 | 818−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8141, large, F81, 11) (dual of [large, large−41, 12]-code), using
- 1 times m-reduction [i] based on digital (30, 41, large)-net over F81, using
(32, 32+8, large)-Net in Base 81 — Upper bound on s
There is no (32, 40, large)-net in base 81, because
- 6 times m-reduction [i] would yield (32, 34, large)-net in base 81, but