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A new treatment of Bose-Chaudhuri codes. (English) Zbl 0137.13604
Summary: Letting \(A\) be any \((k,n)\) Bose-Chaudhuri code, we first attach to each \(a\) in \(A\) (via difference equations over \(\mathrm{GF}(2)\)) a polynomial \(g_a (x)\) such that the coordinates of \(a\) are the values of \(g_a(x)\) on the \(n\)th roots of unity. The degree of these polynomials is such that the minimum nonzero weight \(d\) of vectors in \(A\) is immediately seen to be at least \(d_0\), the usual Bose-Chaudhuri lower bound. This lower bound \(d_0\) is improved over a class of \((h + 1,p)\) codes, where \(p = 2h + 1\) has certain prime values, in a number of general theorems. In particular, the \((12, 23)\) Golay code is proved very simply to have \(d = 7\); and a \((24, 47)\) code is shown to have \(d\leq 9\), thus improving by 4 the usual lower bound \(d_0 = 5\) for that code.

MSC:
94B05 Linear codes, general
94B65 Bounds on codes
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