Best Known (27, 27+7, s)-Nets in Base 32
(27, 27+7, 2796233)-Net over F32 — Constructive and digital
Digital (27, 34, 2796233)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (0, 3, 33)-net over F32, using
- net from sequence [i] based on digital (0, 32)-sequence over F32, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 0 and N(F) ≥ 33, using
- the rational function field F32(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 32)-sequence over F32, using
- digital (24, 31, 2796200)-net over F32, using
- net defined by OOA [i] based on linear OOA(3231, 2796200, F32, 7, 7) (dual of [(2796200, 7), 19573369, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(3231, 8388601, F32, 7) (dual of [8388601, 8388570, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(3231, large, F32, 7) (dual of [large, large−31, 8]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 33554433 | 3210−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(3231, large, F32, 7) (dual of [large, large−31, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(3231, 8388601, F32, 7) (dual of [8388601, 8388570, 8]-code), using
- net defined by OOA [i] based on linear OOA(3231, 2796200, F32, 7, 7) (dual of [(2796200, 7), 19573369, 8]-NRT-code), using
- digital (0, 3, 33)-net over F32, using
(27, 27+7, 2797225)-Net in Base 32 — Constructive
(27, 34, 2797225)-net in base 32, using
- (u, u+v)-construction [i] based on
- digital (1, 4, 1025)-net over F32, using
- net defined by OOA [i] based on linear OOA(324, 1025, F32, 3, 3) (dual of [(1025, 3), 3071, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(324, 1025, F32, 2, 3) (dual of [(1025, 2), 2046, 4]-NRT-code), using
- net defined by OOA [i] based on linear OOA(324, 1025, F32, 3, 3) (dual of [(1025, 3), 3071, 4]-NRT-code), using
- (23, 30, 2796200)-net in base 32, using
- base change [i] based on digital (18, 25, 2796200)-net over F64, using
- net defined by OOA [i] based on linear OOA(6425, 2796200, F64, 7, 7) (dual of [(2796200, 7), 19573375, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(6425, 8388601, F64, 7) (dual of [8388601, 8388576, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(6425, large, F64, 7) (dual of [large, large−25, 8]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 648−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(6425, large, F64, 7) (dual of [large, large−25, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(6425, 8388601, F64, 7) (dual of [8388601, 8388576, 8]-code), using
- net defined by OOA [i] based on linear OOA(6425, 2796200, F64, 7, 7) (dual of [(2796200, 7), 19573375, 8]-NRT-code), using
- base change [i] based on digital (18, 25, 2796200)-net over F64, using
- digital (1, 4, 1025)-net over F32, using
(27, 27+7, large)-Net over F32 — Digital
Digital (27, 34, large)-net over F32, using
- 323 times duplication [i] based on digital (24, 31, large)-net over F32, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3231, large, F32, 7) (dual of [large, large−31, 8]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 33554433 | 3210−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3231, large, F32, 7) (dual of [large, large−31, 8]-code), using
(27, 27+7, large)-Net in Base 32 — Upper bound on s
There is no (27, 34, large)-net in base 32, because
- 5 times m-reduction [i] would yield (27, 29, large)-net in base 32, but