Best Known (70−15, 70, s)-Nets in Base 32
(70−15, 70, 150139)-Net over F32 — Constructive and digital
Digital (55, 70, 150139)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (6, 13, 342)-net over F32, using
- net defined by OOA [i] based on linear OOA(3213, 342, F32, 7, 7) (dual of [(342, 7), 2381, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(3213, 1027, F32, 7) (dual of [1027, 1014, 8]-code), using
- construction XX applied to C1 = C([1022,4]), C2 = C([0,5]), C3 = C1 + C2 = C([0,4]), and C∩ = C1 ∩ C2 = C([1022,5]) [i] based on
- linear OA(3211, 1023, F32, 6) (dual of [1023, 1012, 7]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−1,0,…,4}, and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(3211, 1023, F32, 6) (dual of [1023, 1012, 7]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(3213, 1023, F32, 7) (dual of [1023, 1010, 8]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−1,0,…,5}, and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(329, 1023, F32, 5) (dual of [1023, 1014, 6]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,4], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(320, 2, F32, 0) (dual of [2, 2, 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
- linear OA(320, 2, F32, 0) (dual of [2, 2, 1]-code) (see above)
- construction XX applied to C1 = C([1022,4]), C2 = C([0,5]), C3 = C1 + C2 = C([0,4]), and C∩ = C1 ∩ C2 = C([1022,5]) [i] based on
- OOA 3-folding and stacking with additional row [i] based on linear OA(3213, 1027, F32, 7) (dual of [1027, 1014, 8]-code), using
- net defined by OOA [i] based on linear OOA(3213, 342, F32, 7, 7) (dual of [(342, 7), 2381, 8]-NRT-code), using
- digital (42, 57, 149797)-net over F32, using
- net defined by OOA [i] based on linear OOA(3257, 149797, F32, 15, 15) (dual of [(149797, 15), 2246898, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(3257, 1048580, F32, 15) (dual of [1048580, 1048523, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(13) [i] based on
- linear OA(3257, 1048576, F32, 15) (dual of [1048576, 1048519, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(3253, 1048576, F32, 14) (dual of [1048576, 1048523, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [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 (see above)
- construction X applied to Ce(14) ⊂ Ce(13) [i] based on
- OOA 7-folding and stacking with additional row [i] based on linear OA(3257, 1048580, F32, 15) (dual of [1048580, 1048523, 16]-code), using
- net defined by OOA [i] based on linear OOA(3257, 149797, F32, 15, 15) (dual of [(149797, 15), 2246898, 16]-NRT-code), using
- digital (6, 13, 342)-net over F32, using
(70−15, 70, 1198371)-Net in Base 32 — Constructive
(55, 70, 1198371)-net in base 32, using
- 321 times duplication [i] based on (54, 69, 1198371)-net in base 32, using
- net defined by OOA [i] based on OOA(3269, 1198371, S32, 15, 15), using
- OOA 7-folding and stacking with additional row [i] based on OA(3269, 8388598, S32, 15), using
- discarding factors based on OA(3269, large, S32, 15), using
- discarding parts of the base [i] based on linear OA(6457, large, F64, 15) (dual of [large, large−57, 16]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 648−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- discarding parts of the base [i] based on linear OA(6457, large, F64, 15) (dual of [large, large−57, 16]-code), using
- discarding factors based on OA(3269, large, S32, 15), using
- OOA 7-folding and stacking with additional row [i] based on OA(3269, 8388598, S32, 15), using
- net defined by OOA [i] based on OOA(3269, 1198371, S32, 15, 15), using
(70−15, 70, 6544047)-Net over F32 — Digital
Digital (55, 70, 6544047)-net over F32, using
(70−15, 70, large)-Net in Base 32 — Upper bound on s
There is no (55, 70, large)-net in base 32, because
- 13 times m-reduction [i] would yield (55, 57, large)-net in base 32, but