A characterization of flat spaces in a finite geometry and the uniqueness of the hamming and the MacDonald codes
A characterization of flat spaces in a finite geometry and the uniqueness of the hamming and the MacDonald codes
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DOI：
10.1016/s0021-9800(66)80007-8
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发表时间：
1966-06
期刊：
影响因子：
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通讯作者：
R. C. Bose;R. Burton
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文献类型：
--
作者：
R. C. Bose;R. Burton
设PG（k−1，q）是有限域GF（q）上k −1维的射影几何，其中q是素数幂。PG（k−1，q）的平坦空间可以由以下定理来刻画。如果F是PG（k−1，q）中的一个点集，它与每个v-平坦空间都有一个非空交，则F中的点数大于或等于（qk−v−1）/（q−1）。等式成立当且仅当F是（k−v，−1）-平坦空间。从这个定理可以证明，对于给定的冗余度r，q=2和最小距离=4是唯一的。该定理的一个推广证明了d =qk−1−qu（u=0，1，...，k−2）的MacDonald码是唯一的。
LetPG(k−1, q)be the projective geometry of dimensionk−1 over the finite fieldGF(q)whereqis a prime power. The flat spaces ofPG(k−1, q)may be characterized by the followingTheorem.If F is a set of points in PG(k−1,q)which has a non-empty intersection with every v-flat, then the number of points in F is greater than or equal to (qk−v−1)/(q−1).Equality holds if, and only if, F is a(k−v, −1)-flat.It may be shown from this theorem that the Hamming codes which maximizenfor a given redundancyr, q=2, and minimum distanced=4, are unique. An extension of the theorem shows that the MacDonald codes withd=qk−1−qu(u=0, 1, …,k−2) are unique.