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Does Reed-Muller codes $\mathrm{RM}(r,m)$ exist for $r = 0$ and $m = 0$? Namely, does $\mathrm{RM}(0,0)$ exist? Some books mention that $m$ should be a natural number whereas some books mention that it can be a whole number including $0$, for example Shu Lin and Daniel Costello Error Control Coding book.

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  • $\begingroup$ When r and are non zero, also it is said that r can be equal to m. In such a case n = k = 2 raised to m. How can the message length be equal to codeword length? Is it that r should be less than m? k is given as mC0+mC1+.....mCr. And n is given as 2 raised to m. $\endgroup$ – Seetha Rama Raju Sanapala Feb 24 '16 at 10:53
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The codewords in a Reed-Muller code of degree $d$ and length $2^n$ are binary vectors of the form $$\big(f(0,0,\ldots, 0, 0), f(0,0, \ldots, 0, 1), f(0,0, \ldots, 1, 0), f(0,0, \ldots, 1, 1), \cdots, f(1,1,\ldots, 1,1) \big)$$ where $f$ is the corresponding binary polynomial of degree at most $d$ in $n$ variables.

Suppose first that $n > 0$. The dimension of the code is $$k = \sum_{i=0}^d \binom{n}{i}\tag{1}$$ If $d=0$, then $k = \binom{n}{0}=1.$ There are only two "polynomials" of degree $0$ in $n$ variables, namely the constants $0$ and $1$, and consequently the codewords of length $2^n$ are $000\cdots 0$ and $111\cdots 1$.

If $n=0$ also, the codewords (if any) are of length $2^0 = 1$. Now, you might want to claim from $(1)$ that $k = \binom{0}{0} = \frac{0!}{0!(0-0)!} = 1$ and so there are $2^k = 2$ codewords (which, of course, are $0$ and $1$). Alternatively, the $k$ in Equation $(1)$ is the total number of subsets of cardinality $d$ or less of a set of $n$ elements. There is exactly one subset (of cardinality $0$) of the set of $0$ elements (namely,the empty set $\emptyset$). and that subset is $\emptyset$ itself. Thus, the Reed-Muller code of order $0$ and length $2^0$ is the $(1,1)$ code consisting of two codewords $0$ and $1$. As a linear code, this is the identity map.

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