Generator Matrices and Symmetrized Weight Enumerators of Linear Codes over <inline-formula><math display="inline"><semantics><mrow><mi mathvariant="double-struck">F</mi></mrow></semantics></math></inline-formula><i>_p^m</i> + <i>u</i><inline-formula><math display="inline"><semantics><mrow><mi mathvariant="double-struck">F</mi></mrow></semantics></math></inline-formula><i>_p^m</i> + <i>v</i><inline-formula><math display="inline"><semantics><mrow><mi mathvariant="double-struck">F</mi></mrow></semantics></math></inline-formula><i>_p^m</i> + <i>w</i><inline-formula><math display="inline"><semantics><mrow><mi mathvariant="double-struck">F</mi></mrow></semantics></math></inline-formula><i>_p^m</i>

Abstract

Let u,v, and w be indeterminates over Fpm and let R=Fpm+uFpm+vFpm+wFpm, where p is a prime. Then, R is a ring of order p4m, and R≅Fpm[u,v,w]I with maximal ideal J=uFpm+vFpm+wFpm of order p3m and a residue field Fpm of order pm, where I is an appropriate ideal. In this article, the goal is to improve the understanding of linear codes over local non-chain rings. In particular, we investigate the symmetrized weight enumerators and generator matrices of linear codes of length N over R. In order to accomplish that, we first list all such rings up to the isomorphism for different values of the index of nilpotency l of J, 2≤l≤4. Furthermore, we fully describe the lattice of ideals of R and their orders. Next, for linear codes C over R, we compute the generator matrices and symmetrized weight enumerators, as shown by numerical examples.

Keywords

• symmetrized weight enumerator
• coding over rings
• Frobenius rings
• generator matrix