Best Known (18, 31, s)-Nets in Base 81
(18, 31, 1175)-Net over F81 — Constructive and digital
Digital (18, 31, 1175)-net over F81, using
- (u, u+v)-construction [i] based on
- digital (0, 6, 82)-net over F81, using
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 0 and N(F) ≥ 82, using
- the rational function field F81(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- digital (12, 25, 1093)-net over F81, using
- net defined by OOA [i] based on linear OOA(8125, 1093, F81, 13, 13) (dual of [(1093, 13), 14184, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(8125, 6559, F81, 13) (dual of [6559, 6534, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(8125, 6561, F81, 13) (dual of [6561, 6536, 14]-code), using
- an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- discarding factors / shortening the dual code based on linear OA(8125, 6561, F81, 13) (dual of [6561, 6536, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(8125, 6559, F81, 13) (dual of [6559, 6534, 14]-code), using
- net defined by OOA [i] based on linear OOA(8125, 1093, F81, 13, 13) (dual of [(1093, 13), 14184, 14]-NRT-code), using
- digital (0, 6, 82)-net over F81, using
(18, 31, 6645)-Net over F81 — Digital
Digital (18, 31, 6645)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8131, 6645, F81, 13) (dual of [6645, 6614, 14]-code), using
- (u, u+v)-construction [i] based on
- linear OA(816, 82, F81, 6) (dual of [82, 76, 7]-code or 82-arc in PG(5,81)), using
- extended Reed–Solomon code RSe(76,81) [i]
- linear OA(8125, 6563, F81, 13) (dual of [6563, 6538, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- linear OA(8125, 6561, F81, 13) (dual of [6561, 6536, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(8123, 6561, F81, 12) (dual of [6561, 6538, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(810, 2, F81, 0) (dual of [2, 2, 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(12) ⊂ Ce(11) [i] based on
- linear OA(816, 82, F81, 6) (dual of [82, 76, 7]-code or 82-arc in PG(5,81)), using
- (u, u+v)-construction [i] based on
(18, 31, large)-Net in Base 81 — Upper bound on s
There is no (18, 31, large)-net in base 81, because
- 11 times m-reduction [i] would yield (18, 20, large)-net in base 81, but