On the Theory of $\mathbb{Z}_{m}$ Linear Codes: the $\mathbb{Z}_{m}$ Linear Preparata and Goethals Codes for $m>2$ *

Let <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$m \in \mathbb{N}$</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathbb{Z}_{m} \stackrel{\text { def }}{=}\{0,1, \ldots, m-1\}$</tex> be the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$m$</tex> ary alphabet. <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathbf{A} \mathbb{Z}_{m}$</tex> linear code of length <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$n \in \mathbb{N}$</tex> is a submodule of the module <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\left(\mathbb{Z}_{m}^{n},+\bmod m, \mathbb{Z}_{m}, \cdot \bmod m\right)$</tex>. This paper presents a significant lower bound for the minimum Lee distance <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$d_{\text {Lee }}(\mathcal{C})$</tex>, of any <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathbb{Z}_{m}$</tex> linear code <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathcal{C}$</tex>. 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 <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathbb{Z}_{4}$</tex> linear Preparata and Goethals codes over the alphabet <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathbb{Z}_{m}, m=2^{l} \in \mathbb{N}$</tex> with <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$l \geq 2$</tex>. Specifically, for any <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$m=2^{l}&gt;2$</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$h \in \mathbb{N}, h$</tex> odd, the generalized <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathbb{Z}_{m}$</tex> linear Preparata codes have length <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$n+1=2^{h}$</tex>, minimum Lee distance 6 and cardinality <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$|\mathcal{C}|=m^{n-h-1}\lceil m / 4\rceil^{h}\lceil m / 8\rceil$</tex>. For the same parameters <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$m, l, n, h \in \mathbb{N}$</tex>, the generalized <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathbb{Z}_{m}$</tex> linear Goethals codes have length <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$n+1=2^{h}$</tex>, minimum Lee distance 8 and cardinality <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$|\mathcal{C}|=m^{n-2 h-1}\lceil m / 2\rceil^{h}\lceil m / 4\rceil^{h}\lceil m / 8\rceil$</tex>. Notably, both families of codes are less redundant and less complex than <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$m$</tex>-ary codes with the same minimum Lee distances obtained by applying the Gray mapping to <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$l$</tex>-bit subblocks of codewords from binary linear codes.