Best Known (60, 60+13, s)-Nets in Base 81
(60, 60+13, 2796201)-Net over F81 — Constructive and digital
Digital (60, 73, 2796201)-net over F81, using
- net defined by OOA [i] based on linear OOA(8173, 2796201, F81, 15, 13) (dual of [(2796201, 15), 41942942, 14]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(8173, 5592403, F81, 3, 13) (dual of [(5592403, 3), 16777136, 14]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(8170, 5592402, F81, 3, 13) (dual of [(5592402, 3), 16777136, 14]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(8121, 2796201, F81, 3, 6) (dual of [(2796201, 3), 8388582, 7]-NRT-code), using
- OOA 3-folding [i] based on linear OA(8121, large, F81, 6) (dual of [large, large−21, 7]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523360 | 814−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [i]
- OOA 3-folding [i] based on linear OA(8121, large, F81, 6) (dual of [large, large−21, 7]-code), using
- linear OOA(8149, 2796201, F81, 3, 13) (dual of [(2796201, 3), 8388554, 14]-NRT-code), using
- OOA 3-folding [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]
- OOA 3-folding [i] based on linear OA(8149, large, F81, 13) (dual of [large, large−49, 14]-code), using
- linear OOA(8121, 2796201, F81, 3, 6) (dual of [(2796201, 3), 8388582, 7]-NRT-code), using
- (u, u+v)-construction [i] based on
- 1 times NRT-code embedding in larger space [i] based on linear OOA(8170, 5592402, F81, 3, 13) (dual of [(5592402, 3), 16777136, 14]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(8173, 5592403, F81, 3, 13) (dual of [(5592403, 3), 16777136, 14]-NRT-code), using
(60, 60+13, large)-Net over F81 — Digital
Digital (60, 73, large)-net over F81, using
- 8 times m-reduction [i] based on digital (60, 81, large)-net over F81, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8181, large, F81, 21) (dual of [large, large−81, 22]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523361 | 818−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8181, large, F81, 21) (dual of [large, large−81, 22]-code), using
(60, 60+13, large)-Net in Base 81 — Upper bound on s
There is no (60, 73, large)-net in base 81, because
- 11 times m-reduction [i] would yield (60, 62, large)-net in base 81, but