Best Known (36, 36+9, s)-Nets in Base 81
(36, 36+9, 2362954)-Net over F81 — Constructive and digital
Digital (36, 45, 2362954)-net over F81, using
- (u, u+v)-construction [i] based on
- digital (8, 12, 265804)-net over F81, using
- net defined by OOA [i] based on linear OOA(8112, 265804, F81, 4, 4) (dual of [(265804, 4), 1063204, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(8112, 265804, F81, 3, 4) (dual of [(265804, 3), 797400, 5]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(812, 82, F81, 3, 2) (dual of [(82, 3), 244, 3]-NRT-code), using
- extended Reed–Solomon NRT-code RSe(3;244,81) [i]
- linear OOA(8110, 265722, F81, 3, 4) (dual of [(265722, 3), 797156, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(8110, 531444, F81, 4) (dual of [531444, 531434, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(2) [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(817, 531441, F81, 3) (dual of [531441, 531434, 4]-code or 531441-cap in PG(6,81)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [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(3) ⊂ Ce(2) [i] based on
- OA 2-folding and stacking [i] based on linear OA(8110, 531444, F81, 4) (dual of [531444, 531434, 5]-code), using
- linear OOA(812, 82, F81, 3, 2) (dual of [(82, 3), 244, 3]-NRT-code), using
- (u, u+v)-construction [i] based on
- appending kth column [i] based on linear OOA(8112, 265804, F81, 3, 4) (dual of [(265804, 3), 797400, 5]-NRT-code), using
- net defined by OOA [i] based on linear OOA(8112, 265804, F81, 4, 4) (dual of [(265804, 4), 1063204, 5]-NRT-code), using
- digital (24, 33, 2097150)-net over F81, using
- net defined by OOA [i] based on linear OOA(8133, 2097150, F81, 9, 9) (dual of [(2097150, 9), 18874317, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(8133, 8388601, F81, 9) (dual of [8388601, 8388568, 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 and stacking with additional row [i] based on linear OA(8133, 8388601, F81, 9) (dual of [8388601, 8388568, 10]-code), using
- net defined by OOA [i] based on linear OOA(8133, 2097150, F81, 9, 9) (dual of [(2097150, 9), 18874317, 10]-NRT-code), using
- digital (8, 12, 265804)-net over F81, using
(36, 36+9, large)-Net over F81 — Digital
Digital (36, 45, large)-net over F81, using
- 4 times m-reduction [i] based on digital (36, 49, large)-net over F81, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8149, large, F81, 13) (dual of [large, large−49, 14]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523361 | 818−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8149, large, F81, 13) (dual of [large, large−49, 14]-code), using
(36, 36+9, large)-Net in Base 81 — Upper bound on s
There is no (36, 45, large)-net in base 81, because
- 7 times m-reduction [i] would yield (36, 38, large)-net in base 81, but