Best Known (61−31, 61, s)-Nets in Base 81
(61−31, 61, 437)-Net over F81 — Constructive and digital
Digital (30, 61, 437)-net over F81, using
- net defined by OOA [i] based on linear OOA(8161, 437, F81, 31, 31) (dual of [(437, 31), 13486, 32]-NRT-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(8161, 6556, F81, 31) (dual of [6556, 6495, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(8161, 6561, F81, 31) (dual of [6561, 6500, 32]-code), using
- an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- discarding factors / shortening the dual code based on linear OA(8161, 6561, F81, 31) (dual of [6561, 6500, 32]-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(8161, 6556, F81, 31) (dual of [6556, 6495, 32]-code), using
(61−31, 61, 1705)-Net over F81 — Digital
Digital (30, 61, 1705)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8161, 1705, F81, 3, 31) (dual of [(1705, 3), 5054, 32]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8161, 2187, F81, 3, 31) (dual of [(2187, 3), 6500, 32]-NRT-code), using
- OOA 3-folding [i] based on linear OA(8161, 6561, F81, 31) (dual of [6561, 6500, 32]-code), using
- an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- OOA 3-folding [i] based on linear OA(8161, 6561, F81, 31) (dual of [6561, 6500, 32]-code), using
- discarding factors / shortening the dual code based on linear OOA(8161, 2187, F81, 3, 31) (dual of [(2187, 3), 6500, 32]-NRT-code), using
(61−31, 61, 3456334)-Net in Base 81 — Upper bound on s
There is no (30, 61, 3456335)-net in base 81, because
- 1 times m-reduction [i] would yield (30, 60, 3456335)-net in base 81, but
- the generalized Rao bound for nets shows that 81m ≥ 3 229246 356357 716823 211875 043660 281183 540924 205623 740181 603585 598532 120840 678139 539629 842012 168765 829187 711471 762001 > 8160 [i]