Best Known (74−13, 74, s)-Nets in Base 32
(74−13, 74, 1398444)-Net over F32 — Constructive and digital
Digital (61, 74, 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 (48, 61, 1398100)-net over F32, using
- net defined by OOA [i] based on linear OOA(3261, 1398100, F32, 13, 13) (dual of [(1398100, 13), 18175239, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(3261, 8388601, F32, 13) (dual of [8388601, 8388540, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(3261, large, F32, 13) (dual of [large, large−61, 14]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 33554433 | 3210−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(3261, large, F32, 13) (dual of [large, large−61, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(3261, 8388601, F32, 13) (dual of [8388601, 8388540, 14]-code), using
- net defined by OOA [i] based on linear OOA(3261, 1398100, F32, 13, 13) (dual of [(1398100, 13), 18175239, 14]-NRT-code), using
- digital (7, 13, 344)-net over F32, using
(74−13, 74, 1399467)-Net in Base 32 — Constructive
(61, 74, 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
- (46, 59, 1398100)-net in base 32, using
- net defined by OOA [i] based on OOA(3259, 1398100, S32, 13, 13), using
- OOA 6-folding and stacking with additional row [i] based on OA(3259, 8388601, S32, 13), using
- discarding factors based on OA(3259, large, S32, 13), using
- discarding parts of the base [i] based on linear OA(6449, large, F64, 13) (dual of [large, large−49, 14]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 648−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- discarding parts of the base [i] based on linear OA(6449, large, F64, 13) (dual of [large, large−49, 14]-code), using
- discarding factors based on OA(3259, large, S32, 13), using
- OOA 6-folding and stacking with additional row [i] based on OA(3259, 8388601, S32, 13), using
- net defined by OOA [i] based on OOA(3259, 1398100, S32, 13, 13), using
- (9, 15, 1367)-net in base 32, using
(74−13, 74, large)-Net over F32 — Digital
Digital (61, 74, large)-net over F32, using
- t-expansion [i] based on digital (60, 74, large)-net over F32, using
- 2 times m-reduction [i] based on digital (60, 76, large)-net over F32, using
(74−13, 74, large)-Net in Base 32 — Upper bound on s
There is no (61, 74, large)-net in base 32, because
- 11 times m-reduction [i] would yield (61, 63, large)-net in base 32, but