Best Known (31, 31+13, s)-Nets in Base 32
(31, 31+13, 5505)-Net over F32 — Constructive and digital
Digital (31, 44, 5505)-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 (24, 37, 5461)-net over F32, using
- net defined by OOA [i] based on linear OOA(3237, 5461, F32, 13, 13) (dual of [(5461, 13), 70956, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(3237, 32767, F32, 13) (dual of [32767, 32730, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(3237, 32768, F32, 13) (dual of [32768, 32731, 14]-code), using
- an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- discarding factors / shortening the dual code based on linear OA(3237, 32768, F32, 13) (dual of [32768, 32731, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(3237, 32767, F32, 13) (dual of [32767, 32730, 14]-code), using
- net defined by OOA [i] based on linear OOA(3237, 5461, F32, 13, 13) (dual of [(5461, 13), 70956, 14]-NRT-code), using
- digital (1, 7, 44)-net over F32, using
(31, 31+13, 10923)-Net in Base 32 — Constructive
(31, 44, 10923)-net in base 32, using
- 323 times duplication [i] based on (28, 41, 10923)-net in base 32, using
- net defined by OOA [i] based on OOA(3241, 10923, S32, 13, 13), using
- OOA 6-folding and stacking with additional row [i] based on OA(3241, 65539, S32, 13), using
- 1 times code embedding in larger space [i] based on OA(3240, 65538, S32, 13), using
- discarding parts of the base [i] based on linear OA(25625, 65538, F256, 13) (dual of [65538, 65513, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- linear OA(25625, 65536, F256, 13) (dual of [65536, 65511, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(25623, 65536, F256, 12) (dual of [65536, 65513, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(2560, 2, F256, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(2560, s, F256, 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(25625, 65538, F256, 13) (dual of [65538, 65513, 14]-code), using
- 1 times code embedding in larger space [i] based on OA(3240, 65538, S32, 13), using
- OOA 6-folding and stacking with additional row [i] based on OA(3241, 65539, S32, 13), using
- net defined by OOA [i] based on OOA(3241, 10923, S32, 13, 13), using
(31, 31+13, 56355)-Net over F32 — Digital
Digital (31, 44, 56355)-net over F32, using
(31, 31+13, large)-Net in Base 32 — Upper bound on s
There is no (31, 44, large)-net in base 32, because
- 11 times m-reduction [i] would yield (31, 33, large)-net in base 32, but