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The concept “perfect code” was introduced in (2.2.3) and then in (2.2.4) it was shown that Hamming codes over GF(q) are perfect. We recall that these codes qm 1 were defined as follows: Let H be the m by n := \( \frac{{q^m - 1}} {{q - 1}} \) matrix consisting of all the different nonzero columns with elements from GF(q) such that the first nonzero entry in each column is 1. Since no linear combination of two columns can be o we see that H is the parity check matrix of a linear (n,k) code over GF(q) with minimum distance 3. Because qk{1 + n(q-1)} = qm+k = qn the code is perfect.
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© 1971 Springer-Verlag Berlin · Heidelberg
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(1971). Perfect codes. In: Coding Theory. Lecture Notes in Mathematics, vol 201. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-36657-7_5
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DOI: https://doi.org/10.1007/978-3-540-36657-7_5
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-06363-6
Online ISBN: 978-3-540-36657-7
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