Best Known (42, 42+12, s)-Nets in Base 32
(42, 42+12, 174829)-Net over F32 — Constructive and digital
Digital (42, 54, 174829)-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 (33, 45, 174763)-net over F32, using
- net defined by OOA [i] based on linear OOA(3245, 174763, F32, 12, 12) (dual of [(174763, 12), 2097111, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(3245, 1048578, F32, 12) (dual of [1048578, 1048533, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(3245, 1048580, F32, 12) (dual of [1048580, 1048535, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- linear OA(3245, 1048576, F32, 12) (dual of [1048576, 1048531, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(3241, 1048576, F32, 11) (dual of [1048576, 1048535, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(320, 4, F32, 0) (dual of [4, 4, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(320, s, F32, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(3245, 1048580, F32, 12) (dual of [1048580, 1048535, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(3245, 1048578, F32, 12) (dual of [1048578, 1048533, 13]-code), using
- net defined by OOA [i] based on linear OOA(3245, 174763, F32, 12, 12) (dual of [(174763, 12), 2097111, 13]-NRT-code), using
- digital (3, 9, 66)-net over F32, using
(42, 42+12, 1398100)-Net in Base 32 — Constructive
(42, 54, 1398100)-net in base 32, using
- base change [i] based on digital (33, 45, 1398100)-net over F64, using
- net defined by OOA [i] based on linear OOA(6445, 1398100, F64, 12, 12) (dual of [(1398100, 12), 16777155, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(6445, 8388600, F64, 12) (dual of [8388600, 8388555, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(6445, large, F64, 12) (dual of [large, large−45, 13]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 644−1, defining interval I = [0,11], and designed minimum distance d ≥ |I|+1 = 13 [i]
- discarding factors / shortening the dual code based on linear OA(6445, large, F64, 12) (dual of [large, large−45, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(6445, 8388600, F64, 12) (dual of [8388600, 8388555, 13]-code), using
- net defined by OOA [i] based on linear OOA(6445, 1398100, F64, 12, 12) (dual of [(1398100, 12), 16777155, 13]-NRT-code), using
(42, 42+12, 3877673)-Net over F32 — Digital
Digital (42, 54, 3877673)-net over F32, using
(42, 42+12, 6365424)-Net in Base 32
(42, 54, 6365424)-net in base 32, using
- base change [i] based on digital (33, 45, 6365424)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6445, 6365424, F64, 12) (dual of [6365424, 6365379, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(6445, large, F64, 12) (dual of [large, large−45, 13]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 644−1, defining interval I = [0,11], and designed minimum distance d ≥ |I|+1 = 13 [i]
- discarding factors / shortening the dual code based on linear OA(6445, large, F64, 12) (dual of [large, large−45, 13]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6445, 6365424, F64, 12) (dual of [6365424, 6365379, 13]-code), using
(42, 42+12, large)-Net in Base 32 — Upper bound on s
There is no (42, 54, large)-net in base 32, because
- 10 times m-reduction [i] would yield (42, 44, large)-net in base 32, but