Geodesic
Pascal matrices, Reed--Muller expressions and Reed--Muller error correcting codes
Zbornik radova, Tome 18 (2015) no. 26, p. 145

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Reed--Muller expressions are a way to analytically describe binary sequences viewed as truth-vectors of switching functions. Reed--Muller codes are a mean to improve reliability in transmitting binary sequences. Both concepts can be defined in terms of Reed--Muller matrices. These matrices can be viewed as a Pascal matrix (matrix of binomial coefficients) computed modulo $2$. The Pascal matrix is a key concept starting from which these two concepts, the Reed--Muller expressions and the Reed--Muller codes, can be developed. Further, the Pascal matrices give a way towards two different, but equally possible, interpretations of the Reed--Muller expressions. Their elements are coefficients in polynomial expressions, while their columns can be viewed as basis functions in particular spectral representations. Particular rows of the Pascal matrix modulo $2$ are selected to form the generator matrix of the Reed--Muller code. The link to Pascal matrices permits generalizations of the binary Reed--Muller codes into $p$-ary Reed--Muller codes. This link can be used conversely to define new functional expressions for $p$-valued functions.
Classification : 15-02, 43A32 94B05
Keywords: Reed-Muler expressions, Reed-Muler codes, Pascal matrices, spectral transforms 
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     author = {Radomir S. Stankovi\'c and Jaakko Astola and Claudio Moraga},
     title = {Pascal matrices, {Reed--Muller} expressions and {Reed--Muller} error correcting codes},
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Radomir S. Stanković; Jaakko Astola; Claudio Moraga. Pascal matrices, Reed--Muller expressions and Reed--Muller error correcting codes. Zbornik radova, Tome 18 (2015) no. 26, p. 145 . http://geodesic.mathdoc.fr/item/ZR_2015_18_26_a5/