Best Known (3, 7, s)-Nets in Base 32
(3, 7, 513)-Net over F32 — Constructive and digital
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
(3, 7, 1027)-Net over F32 — Digital
Digital (3, 7, 1027)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(327, 1027, F32, 4) (dual of [1027, 1020, 5]-code), using
- construction XX applied to C1 = C([1022,1]), C2 = C([0,2]), C3 = C1 + C2 = C([0,1]), and C∩ = C1 ∩ C2 = C([1022,2]) [i] based on
- linear OA(325, 1023, F32, 3) (dual of [1023, 1018, 4]-code or 1023-cap in PG(4,32)), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−1,0,1}, and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(325, 1023, F32, 3) (dual of [1023, 1018, 4]-code or 1023-cap in PG(4,32)), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,2], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(327, 1023, F32, 4) (dual of [1023, 1016, 5]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−1,0,1,2}, and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(323, 1023, F32, 2) (dual of [1023, 1020, 3]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,1], 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
- linear OA(320, 2, F32, 0) (dual of [2, 2, 1]-code) (see above)
- construction XX applied to C1 = C([1022,1]), C2 = C([0,2]), C3 = C1 + C2 = C([0,1]), and C∩ = C1 ∩ C2 = C([1022,2]) [i] based on
(3, 7, 8455)-Net in Base 32 — Upper bound on s
There is no (3, 7, 8456)-net in base 32, because
- the generalized Rao bound for nets shows that 32m ≥ 34362 228629 > 327 [i]