Best Known (31, 31+32, s)-Nets in Base 81
(31, 31+32, 410)-Net over F81 — Constructive and digital
Digital (31, 63, 410)-net over F81, using
- net defined by OOA [i] based on linear OOA(8163, 410, F81, 32, 32) (dual of [(410, 32), 13057, 33]-NRT-code), using
- OA 16-folding and stacking [i] based on linear OA(8163, 6560, F81, 32) (dual of [6560, 6497, 33]-code), using
- discarding factors / shortening the dual code based on linear OA(8163, 6561, F81, 32) (dual of [6561, 6498, 33]-code), using
- an extension Ce(31) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,31], and designed minimum distance d ≥ |I|+1 = 32 [i]
- discarding factors / shortening the dual code based on linear OA(8163, 6561, F81, 32) (dual of [6561, 6498, 33]-code), using
- OA 16-folding and stacking [i] based on linear OA(8163, 6560, F81, 32) (dual of [6560, 6497, 33]-code), using
(31, 31+32, 1723)-Net over F81 — Digital
Digital (31, 63, 1723)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8163, 1723, F81, 3, 32) (dual of [(1723, 3), 5106, 33]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8163, 2187, F81, 3, 32) (dual of [(2187, 3), 6498, 33]-NRT-code), using
- OOA 3-folding [i] based on linear OA(8163, 6561, F81, 32) (dual of [6561, 6498, 33]-code), using
- an extension Ce(31) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,31], and designed minimum distance d ≥ |I|+1 = 32 [i]
- OOA 3-folding [i] based on linear OA(8163, 6561, F81, 32) (dual of [6561, 6498, 33]-code), using
- discarding factors / shortening the dual code based on linear OOA(8163, 2187, F81, 3, 32) (dual of [(2187, 3), 6498, 33]-NRT-code), using
(31, 31+32, 2780400)-Net in Base 81 — Upper bound on s
There is no (31, 63, 2780401)-net in base 81, because
- the generalized Rao bound for nets shows that 81m ≥ 1 716162 540904 510763 719618 353961 258422 047458 505737 933898 051492 387981 106796 500341 716142 766644 523497 677896 513858 979948 801281 > 8163 [i]