Best Known (31, 31+12, s)-Nets in Base 32
(31, 31+12, 5527)-Net over F32 — Constructive and digital
Digital (31, 43, 5527)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (3, 9, 66)-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 (0, 6, 33)-net over F32, using
- net from sequence [i] based on digital (0, 32)-sequence over F32 (see above)
- digital (0, 3, 33)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (22, 34, 5461)-net over F32, using
- net defined by OOA [i] based on linear OOA(3234, 5461, F32, 12, 12) (dual of [(5461, 12), 65498, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(3234, 32766, F32, 12) (dual of [32766, 32732, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(3234, 32768, F32, 12) (dual of [32768, 32734, 13]-code), using
- an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- discarding factors / shortening the dual code based on linear OA(3234, 32768, F32, 12) (dual of [32768, 32734, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(3234, 32766, F32, 12) (dual of [32766, 32732, 13]-code), using
- net defined by OOA [i] based on linear OOA(3234, 5461, F32, 12, 12) (dual of [(5461, 12), 65498, 13]-NRT-code), using
- digital (3, 9, 66)-net over F32, using
(31, 31+12, 43692)-Net in Base 32 — Constructive
(31, 43, 43692)-net in base 32, using
- net defined by OOA [i] based on OOA(3243, 43692, S32, 12, 12), using
- OA 6-folding and stacking [i] based on OA(3243, 262152, S32, 12), using
- 1 times code embedding in larger space [i] based on OA(3242, 262151, S32, 12), using
- discarding parts of the base [i] based on linear OA(6435, 262151, F64, 12) (dual of [262151, 262116, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(9) [i] based on
- 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(6428, 262144, F64, 10) (dual of [262144, 262116, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(641, 7, F64, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(641, s, F64, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(11) ⊂ Ce(9) [i] based on
- discarding parts of the base [i] based on linear OA(6435, 262151, F64, 12) (dual of [262151, 262116, 13]-code), using
- 1 times code embedding in larger space [i] based on OA(3242, 262151, S32, 12), using
- OA 6-folding and stacking [i] based on OA(3243, 262152, S32, 12), using
(31, 31+12, 121183)-Net over F32 — Digital
Digital (31, 43, 121183)-net over F32, using
(31, 31+12, 131075)-Net in Base 32
(31, 43, 131075)-net in base 32, using
- 321 times duplication [i] based on (30, 42, 131075)-net in base 32, using
- base change [i] based on digital (23, 35, 131075)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6435, 131075, F64, 2, 12) (dual of [(131075, 2), 262115, 13]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6435, 262150, F64, 12) (dual of [262150, 262115, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(6435, 262151, F64, 12) (dual of [262151, 262116, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(9) [i] based on
- 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(6428, 262144, F64, 10) (dual of [262144, 262116, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(641, 7, F64, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(641, s, F64, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(11) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(6435, 262151, F64, 12) (dual of [262151, 262116, 13]-code), using
- OOA 2-folding [i] based on linear OA(6435, 262150, F64, 12) (dual of [262150, 262115, 13]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6435, 131075, F64, 2, 12) (dual of [(131075, 2), 262115, 13]-NRT-code), using
- base change [i] based on digital (23, 35, 131075)-net over F64, using
(31, 31+12, large)-Net in Base 32 — Upper bound on s
There is no (31, 43, large)-net in base 32, because
- 10 times m-reduction [i] would yield (31, 33, large)-net in base 32, but