On the Theory of Zm Linear Codes: the Zm Linear Preparata and Goethals Codes for m>2

Let m ∈ N and ℤm def/ ={0,1, ⋯, m-1} be the m ary alphabet. A ℤm linear code of length n ∈ N is a submodule of the module (ℤmn,+mod m, ℤm, · bmod m). This paper presents a significant lower bound for the minimum Lee distance dLee (C), of any ℤm linear code C. This bound facilitates, for a given minimum Lee distance, the efficient design of high information rate codes, which are computationally simple to implement using algebraic operations over fields of small cardinality. Two notable examples of such codes are provided, demonstrating the application of this bound. These families of codes generalize the Z4 linear Preparata and Goethals codes over the alphabet ℤm, m=2l ∈ N with l ≥ 2. Specifically, for any m=2l>2 and h ∈ N, h odd, the generalized ℤm linear Preparata codes have length n+1=2h, minimum Lee distance 6 and cardinality |C|=mn-h-1⌈ m / 4⌉h⌈ m / 8⌉. For the same parameters m, l, n, h ∈ N, the generalized ℤm linear Goethals codes have length n+1=2h, minimum Lee distance 8 and cardinality |C|=mn-2 h-1⌈ m / 2⌉h⌈ m / 4⌉h⌈ m / 8⌉. Notably, both families of codes are less redundant and less complex than m-ary codes with the same minimum Lee distances obtained by applying the Gray mapping to l-bit subblocks of codewords from binary linear codes.