fact verifications from web
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A short note by Phil dated 1.26.13, written to find web support for some doubtful facts in his earlier Galois material, whose origins he no longer remembers. The text shown covers Fact 4: the minimum distance d of a code equals the minimum number of linearly dependent columns of the parity check matrix H. He cites a Toronto course notes PDF and agrees that a zero column gives d = 1.
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Verification of my Galois Facts PhL 1.26.13
I wrote this so long ago, I have no idea where I got these Facts from. The folder of 1991 shows a WP scan and a Rhee scan, but they are pretty small. Some Facts I don't worry about, but some I do. I will try to accumulate here web support for some of my more dubious Facts.
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Fact 4: The minimum distance d of a code is equal to the minimum number of columns of the matrix H which are linearly dependent. (7.35)
www.cs.toronto.edu/~radford/csc310.S04/week11a.pdf
If a column is all zeros, then you can write α = 0 and in this one column would be called linearly dependent by my definition. I say : For example, three columns C1, C3 and C6 are linearly dependent if αC1 + βC3+ γC6 = 0 for some non-zero α,β,γ. So if only one term, then you have your d = 1.
So I agree: if H has a column of all zeros, then d = 1.
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