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In Chapter LINEAR CODES OVER FINITE RINGS basic functions for working with codes over a
finite ring are described. Because of the large amount of machinery
developed specifically for Z4-codes, this chapter will be devoted
to this special case. The functionality available for Z4-codes consists
of the machinery described in chapter LINEAR CODES OVER FINITE RINGS together with the
contents of this chapter.
This chapter includes constructions for some families of codes over Z4
(see sections Families of Codes over Z4 and New Codes from Old),
efficient functions for computing the rank and dimension of the kernel of
any code over Z4 (Section Structures Associated with the Gray Map), as well as general
functions for computing coset representatives for a subcode in a code over
Z4 (Section Coset Representatives). In addition, there are functions for
computing the permutation automorphism group for Hadamard and extended
perfect codes over Z4, and their cardinal (Section Automorphism Groups).
Finally, various algorithms for decoding codes over Z4 are also provided
(Section Decoding).
Error correcting codes over Z4 are often referred to as
quaternary codes. Important concepts when discussing quaternary
codes are Lee weight and the Gray map, which maps linear
codes over Z4 to (possibly non-linear) codes over Z2.
Many good non-linear binary codes can be defined as the images of
simple linear quaternary codes.
A code over Z4 is a subgroup of Z4n, so it is
isomorphic to an abelian structure Z2γ x Z4δ and we will
say that it is of type 2γ4δ, or simply that it has 2γ + 2δ
codewords. As general references on the available functions in Magma for
codes over Z4, the reader is referred to [HKC+94], [Wan97].
For general references on the material in this chapter, the reader is referred
to the bibliography included at the end of this chapter.
The machinery described in this chapter largely corresponds to Version
2.0 of the package Codes over Z4: A Magma{ Package} which has
been developed by Roland D. Barrolleta, Jaume Pernas, Jaume Pujol and
Mercè Villanueva of the Combinatoric, Coding and Security Group (CCSG)
at the Universitat Autònoma de Barcelona.
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