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By Frederick G. Bohme, United States. Bureau of the Census

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Xn ∈ Fm . Show that: 1) The linear map : Lm → Fn such that (f ) = (f (x1 ), . . , f (xn )) is injective. 2) The image of is a linear code of type [n, m + 1, n − q m−1 ]. 54 Chapter 1. Block Error-correcting Codes Such codes are called (first order) Reed–Muller codes and will be denoted m x RM x q (m). In the case n = q , instead of RM q (m) we will simply write RM q (m) and we will say that this is a full Reed–Muller code. Thus RM q (m) ∼ (q m , m + 1, q m−1 (q − 1)). 47 Remark. There are many references that present results on the decoding of RM codes.

33. For i = 1, . . , αr ), with αi+1 , . . , αn ∈ F arbitrary. Let H = H1 |H2 | · · · |Hr . Prove that H is a q-ary Hamming matrix of codimension r (we will say that it is a normalized q-ary Hamming matrix of codimension r). Note also that Hi has q r−i columns and hence that H has (q r − 1)/(q − 1) columns. The link 19 Normalized Hamming matrix defines a function based on this procedure, and contains some examples. 39 Remark. Since two vectors of Fr are linearly independent if and only if they are non-zero and define distinct points of the projective space P(Fr ), we see that the number n of columns of a q-ary Hamming matrix of codimension qr − 1 r coincides with the cardinal of P(Fr ), which is n = .

The minimum weight of C coincides with the minimum distance of C: dC = wC . Proof: Let x, y ∈ C. If C is linear, then x − y ∈ C and we have hd (x, y) = |x − y|. Since x − y = 0 if and only if x = y, we see that any distance between distinct elements is the weight of a nonzero vector. Conversely, since |x| = hd (x, 0), and 0 ∈ C because C is linear, the weight of any nonzero vector is the distance between two distinct elements of C. Now the claimed equality follows from the definitions of dC and wC .

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200 years of U.S. census taking: population and housing questions, 1790-1990 by Frederick G. Bohme, United States. Bureau of the Census


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