Best Known (77, 77+12, s)-Nets in Base 32
(77, 77+12, 2796713)-Net over F32 — Constructive and digital
Digital (77, 89, 2796713)-net over F32, using
- generalized (u, u+v)-construction [i] based on
- digital (3, 7, 513)-net over F32, using
- net defined by OOA [i] based on linear OOA(327, 513, F32, 4, 4) (dual of [(513, 4), 2045, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(327, 1026, F32, 4) (dual of [1026, 1019, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- linear OA(327, 1024, F32, 4) (dual of [1024, 1017, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(325, 1024, F32, 3) (dual of [1024, 1019, 4]-code or 1024-cap in PG(4,32)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [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
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- OA 2-folding and stacking [i] based on linear OA(327, 1026, F32, 4) (dual of [1026, 1019, 5]-code), using
- net defined by OOA [i] based on linear OOA(327, 513, F32, 4, 4) (dual of [(513, 4), 2045, 5]-NRT-code), using
- digital (20, 26, 1398100)-net over F32, using
- s-reduction based on digital (20, 26, 2796201)-net over F32, using
- net defined by OOA [i] based on linear OOA(3226, 2796201, F32, 6, 6) (dual of [(2796201, 6), 16777180, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(3226, large, F32, 6) (dual of [large, large−26, 7]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 33554431 = 325−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [i]
- OA 3-folding and stacking [i] based on linear OA(3226, large, F32, 6) (dual of [large, large−26, 7]-code), using
- net defined by OOA [i] based on linear OOA(3226, 2796201, F32, 6, 6) (dual of [(2796201, 6), 16777180, 7]-NRT-code), using
- s-reduction based on digital (20, 26, 2796201)-net over F32, using
- digital (44, 56, 1398100)-net over F32, using
- net defined by OOA [i] based on linear OOA(3256, 1398100, F32, 12, 12) (dual of [(1398100, 12), 16777144, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(3256, 8388600, F32, 12) (dual of [8388600, 8388544, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(3256, large, F32, 12) (dual of [large, large−56, 13]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 33554431 = 325−1, defining interval I = [0,11], and designed minimum distance d ≥ |I|+1 = 13 [i]
- discarding factors / shortening the dual code based on linear OA(3256, large, F32, 12) (dual of [large, large−56, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(3256, 8388600, F32, 12) (dual of [8388600, 8388544, 13]-code), using
- net defined by OOA [i] based on linear OOA(3256, 1398100, F32, 12, 12) (dual of [(1398100, 12), 16777144, 13]-NRT-code), using
- digital (3, 7, 513)-net over F32, using
(77, 77+12, large)-Net over F32 — Digital
Digital (77, 89, large)-net over F32, using
- t-expansion [i] based on digital (75, 89, large)-net over F32, using
- 6 times m-reduction [i] based on digital (75, 95, large)-net over F32, using
(77, 77+12, large)-Net in Base 32 — Upper bound on s
There is no (77, 89, large)-net in base 32, because
- 10 times m-reduction [i] would yield (77, 79, large)-net in base 32, but