Best Known (31, 31+7, s)-Nets in Base 32
(31, 31+7, 3846824)-Net over F32 — Constructive and digital
Digital (31, 38, 3846824)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (4, 7, 1050624)-net over F32, using
- net defined by OOA [i] based on linear OOA(327, 1050624, F32, 3, 3) (dual of [(1050624, 3), 3151865, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(327, 1050624, F32, 2, 3) (dual of [(1050624, 2), 2101241, 4]-NRT-code), using
- net defined by OOA [i] based on linear OOA(327, 1050624, F32, 3, 3) (dual of [(1050624, 3), 3151865, 4]-NRT-code), 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 (4, 7, 1050624)-net over F32, using
(31, 31+7, 5592400)-Net in Base 32 — Constructive
(31, 38, 5592400)-net in base 32, using
- (u, u+v)-construction [i] based on
- digital (5, 8, 3214400)-net over F32, using
- net defined by OOA [i] based on linear OOA(328, 3214400, F32, 3, 3) (dual of [(3214400, 3), 9643192, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(328, 3214400, F32, 2, 3) (dual of [(3214400, 2), 6428792, 4]-NRT-code), using
- net defined by OOA [i] based on linear OOA(328, 3214400, F32, 3, 3) (dual of [(3214400, 3), 9643192, 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 (5, 8, 3214400)-net over F32, using
(31, 31+7, large)-Net over F32 — Digital
Digital (31, 38, large)-net over F32, using
- 322 times duplication [i] based on digital (29, 36, large)-net over F32, using
- t-expansion [i] based on digital (28, 36, large)-net over F32, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3236, large, F32, 8) (dual of [large, large−36, 9]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 33554431 = 325−1, defining interval I = [0,7], and designed minimum distance d ≥ |I|+1 = 9 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3236, large, F32, 8) (dual of [large, large−36, 9]-code), using
- t-expansion [i] based on digital (28, 36, large)-net over F32, using
(31, 31+7, large)-Net in Base 32 — Upper bound on s
There is no (31, 38, large)-net in base 32, because
- 5 times m-reduction [i] would yield (31, 33, large)-net in base 32, but