Best Known (36, 36+19, s)-Nets in Base 81
(36, 36+19, 59049)-Net over F81 — Constructive and digital
Digital (36, 55, 59049)-net over F81, using
- net defined by OOA [i] based on linear OOA(8155, 59049, F81, 19, 19) (dual of [(59049, 19), 1121876, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on 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]
- OOA 9-folding and stacking with additional row [i] based on linear OA(8155, 531442, F81, 19) (dual of [531442, 531387, 20]-code), using
(36, 36+19, 178356)-Net over F81 — Digital
Digital (36, 55, 178356)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8155, 178356, F81, 2, 19) (dual of [(178356, 2), 356657, 20]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8155, 265722, F81, 2, 19) (dual of [(265722, 2), 531389, 20]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8155, 531444, F81, 19) (dual of [531444, 531389, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(17) [i] based on
- linear OA(8155, 531441, F81, 19) (dual of [531441, 531386, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(8152, 531441, F81, 18) (dual of [531441, 531389, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [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(18) ⊂ Ce(17) [i] based on
- OOA 2-folding [i] based on linear OA(8155, 531444, F81, 19) (dual of [531444, 531389, 20]-code), using
- discarding factors / shortening the dual code based on linear OOA(8155, 265722, F81, 2, 19) (dual of [(265722, 2), 531389, 20]-NRT-code), using
(36, 36+19, large)-Net in Base 81 — Upper bound on s
There is no (36, 55, large)-net in base 81, because
- 17 times m-reduction [i] would yield (36, 38, large)-net in base 81, but