Best Known (57, 69, s)-Nets in Base 32
(57, 69, 1398444)-Net over F32 — Constructive and digital
Digital (57, 69, 1398444)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (7, 13, 344)-net over F32, using
- net defined by OOA [i] based on linear OOA(3213, 344, F32, 6, 6) (dual of [(344, 6), 2051, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(3213, 1032, F32, 6) (dual of [1032, 1019, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(2) [i] based on
- linear OA(3211, 1024, F32, 6) (dual of [1024, 1013, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [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(322, 8, F32, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,32)), using
- discarding factors / shortening the dual code based on linear OA(322, 32, F32, 2) (dual of [32, 30, 3]-code or 32-arc in PG(1,32)), using
- Reed–Solomon code RS(30,32) [i]
- discarding factors / shortening the dual code based on linear OA(322, 32, F32, 2) (dual of [32, 30, 3]-code or 32-arc in PG(1,32)), using
- construction X applied to Ce(5) ⊂ Ce(2) [i] based on
- OA 3-folding and stacking [i] based on linear OA(3213, 1032, F32, 6) (dual of [1032, 1019, 7]-code), using
- net defined by OOA [i] based on linear OOA(3213, 344, F32, 6, 6) (dual of [(344, 6), 2051, 7]-NRT-code), 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 (7, 13, 344)-net over F32, using
(57, 69, 1399467)-Net in Base 32 — Constructive
(57, 69, 1399467)-net in base 32, using
- (u, u+v)-construction [i] based on
- (9, 15, 1367)-net in base 32, using
- net defined by OOA [i] based on OOA(3215, 1367, S32, 6, 6), using
- OA 3-folding and stacking [i] based on OA(3215, 4101, S32, 6), using
- discarding parts of the base [i] based on linear OA(6412, 4101, F64, 6) (dual of [4101, 4089, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(3) [i] based on
- linear OA(6411, 4096, F64, 6) (dual of [4096, 4085, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(647, 4096, F64, 4) (dual of [4096, 4089, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(641, 5, F64, 1) (dual of [5, 4, 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(5) ⊂ Ce(3) [i] based on
- discarding parts of the base [i] based on linear OA(6412, 4101, F64, 6) (dual of [4101, 4089, 7]-code), using
- OA 3-folding and stacking [i] based on OA(3215, 4101, S32, 6), using
- net defined by OOA [i] based on OOA(3215, 1367, S32, 6, 6), using
- (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
- base change [i] based on digital (33, 45, 1398100)-net over F64, using
- (9, 15, 1367)-net in base 32, using
(57, 69, large)-Net over F32 — Digital
Digital (57, 69, large)-net over F32, using
- t-expansion [i] based on digital (56, 69, large)-net over F32, using
- 2 times m-reduction [i] based on digital (56, 71, large)-net over F32, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3271, large, F32, 15) (dual of [large, large−71, 16]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 33554433 | 3210−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3271, large, F32, 15) (dual of [large, large−71, 16]-code), using
- 2 times m-reduction [i] based on digital (56, 71, large)-net over F32, using
(57, 69, large)-Net in Base 32 — Upper bound on s
There is no (57, 69, large)-net in base 32, because
- 10 times m-reduction [i] would yield (57, 59, large)-net in base 32, but