Linear codes over \({\mathbb{F}_2+u\mathbb{F}_2+v\mathbb{F}_2+uv\mathbb{F}_2}\)
The authors investigate codes over the ring \(F_2+uF_2 + vF_2 + uv F_2\). These codes are a generalization of codes over \(F_2+uF_2\) and like these codes are equipped with a linear Gray map to binary codes. Lee weights are connected to Hamming weights in the image under this map. Ideals are described in the ring and are used to describe the type of a code and to define independence of generators. MacWilliams relations are given for the complete weight enumerator. Additionally, it is proven that the binary Reed-Muller codes \(RM(r,m)\) are all \(F_2+uF_2 + vF_2 + uv F_2\) linear for \(0 \leq r \leq m.\)
scientific article; zbMATH DE number 5675124