A New Generalization of Reed-Muller Codes
Todd D. Vance · Missouri Journal of Mathematical Sciences · 1997
An error-correcting code can be defined as a set of functions mapping $P$, called the set of places, to $A$, called the alphabet. With classical Generalized Reed-Muller Codes, $P$ is an $m$-dimensional vector space $F^m$ over a finite field $F$, and $A$ is just the finite field $F$. Then, $C= \text{ GRM}( u,m)$ is defined to be the set of all functions from $P$ to $A$ which, when represented as a polynomial of minimal degree through Lagrange interpolation, (see [2], for example) has degree less than or equal to $ u$. This procedure can be generalized. $C=A_{ u}$ is taken to be an element of the filtration of some filtered $F$-algebra $B$. $A$ is another $F$-algebra, and $P = \text{HOM}_{ALG}\ (B,A)$. Then, $C=C_{ u}(B,A)$ is the set of elements of $B_{ u}$ viewed as functions from $P$ to $A$ via $b(x) := x(b)$ for $x\in P$ and $b\in B$.