Best Known (13, 27, s)-Nets in Base 81
(13, 27, 937)-Net over F81 — Constructive and digital
Digital (13, 27, 937)-net over F81, using
- net defined by OOA [i] based on linear OOA(8127, 937, F81, 14, 14) (dual of [(937, 14), 13091, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(8127, 6559, F81, 14) (dual of [6559, 6532, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(8127, 6561, F81, 14) (dual of [6561, 6534, 15]-code), using
- an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- discarding factors / shortening the dual code based on linear OA(8127, 6561, F81, 14) (dual of [6561, 6534, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(8127, 6559, F81, 14) (dual of [6559, 6532, 15]-code), using
(13, 27, 2150)-Net over F81 — Digital
Digital (13, 27, 2150)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8127, 2150, F81, 3, 14) (dual of [(2150, 3), 6423, 15]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8127, 2187, F81, 3, 14) (dual of [(2187, 3), 6534, 15]-NRT-code), using
- OOA 3-folding [i] based on linear OA(8127, 6561, F81, 14) (dual of [6561, 6534, 15]-code), using
- an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- OOA 3-folding [i] based on linear OA(8127, 6561, F81, 14) (dual of [6561, 6534, 15]-code), using
- discarding factors / shortening the dual code based on linear OOA(8127, 2187, F81, 3, 14) (dual of [(2187, 3), 6534, 15]-NRT-code), using
(13, 27, 970793)-Net in Base 81 — Upper bound on s
There is no (13, 27, 970794)-net in base 81, because
- the generalized Rao bound for nets shows that 81m ≥ 3381 416173 638027 719908 607777 109910 692667 828927 100641 > 8127 [i]