Best Known (32, 32+13, s)-Nets in Base 32
(32, 32+13, 5506)-Net over F32 — Constructive and digital
Digital (32, 45, 5506)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (1, 7, 44)-net over F32, using
- net from sequence [i] based on digital (1, 43)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 1 and N(F) ≥ 44, using
- net from sequence [i] based on digital (1, 43)-sequence over F32, using
- digital (25, 38, 5462)-net over F32, using
- net defined by OOA [i] based on linear OOA(3238, 5462, F32, 13, 13) (dual of [(5462, 13), 70968, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(3238, 32773, F32, 13) (dual of [32773, 32735, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(3238, 32776, F32, 13) (dual of [32776, 32738, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([0,5]) [i] based on
- linear OA(3237, 32769, F32, 13) (dual of [32769, 32732, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 326−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(3231, 32769, F32, 11) (dual of [32769, 32738, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 326−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(321, 7, F32, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(321, s, F32, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,6]) ⊂ C([0,5]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3238, 32776, F32, 13) (dual of [32776, 32738, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(3238, 32773, F32, 13) (dual of [32773, 32735, 14]-code), using
- net defined by OOA [i] based on linear OOA(3238, 5462, F32, 13, 13) (dual of [(5462, 13), 70968, 14]-NRT-code), using
- digital (1, 7, 44)-net over F32, using
(32, 32+13, 43691)-Net in Base 32 — Constructive
(32, 45, 43691)-net in base 32, using
- net defined by OOA [i] based on OOA(3245, 43691, S32, 13, 13), using
- OOA 6-folding and stacking with additional row [i] based on OA(3245, 262147, S32, 13), using
- discarding parts of the base [i] based on linear OA(6437, 262147, F64, 13) (dual of [262147, 262110, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- linear OA(6437, 262144, F64, 13) (dual of [262144, 262107, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(6434, 262144, F64, 12) (dual of [262144, 262110, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(640, 3, F64, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(640, s, F64, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- discarding parts of the base [i] based on linear OA(6437, 262147, F64, 13) (dual of [262147, 262110, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on OA(3245, 262147, S32, 13), using
(32, 32+13, 75222)-Net over F32 — Digital
Digital (32, 45, 75222)-net over F32, using
(32, 32+13, large)-Net in Base 32 — Upper bound on s
There is no (32, 45, large)-net in base 32, because
- 11 times m-reduction [i] would yield (32, 34, large)-net in base 32, but