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This chapter describes those functions which are applicable to linear codes
over finite rings. Magma currently supports the basic facilities for codes
over integer residue rings and galois rings, including cyclic codes,
constructions, complete weight enumerators and decoding. Much additional
functionality specific to codes defined over Z4, the integers modulo 4,
can be found in Chapter LINEAR CODES OVER THE INTEGER RESIDUE RING Z4.
For modules defined over rings with zero divisors, it is of course not
possible to talk about the concept of dimension (the modules are not free).
But in Magma each code over such a ring has a unique generator
matrix corresponding to the Howell form. The number of rows k in
this unique generator matrix will be called the pseudo-dimension of
the code. It should be noted that this pseudo-dimension is not invariant
between equivalent codes, and so does not provide structural information
like the dimension of a code over a finite field.
Note that the rank of the generator matrix is always well-defined and
unique (based on the Smith form which is well-defined over PIRs), but k
may sometimes be larger than the rank.
Without a concept of dimension, codes over finite rings are referenced
by their cardinality. A code C is called an (n, M, d) code
if it has length n, cardinality M and minimum Hamming weight d.
The reader is referred to [Wan97] as a general reference on codes
over Galois rings, especially linear codes over Z4.
In this chapter, as for codes over finite fields, the
term "code" will refer to a linear code, unless otherwise specified.
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