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.