Type I and type II codes over the ring \({\mathbb F}_2+u{\mathbb F}_2+u^2{\mathbb F}_2+u^3{\mathbb F}_2\)

<p>We consider a ring <span class="math inline">\(R_{u^3} = \mathbb{F}_2+u\mathbb{F}_2+u^2\mathbb{F}_2+u^3\mathbb{F}_2, u^4=0\)</span>. We discuss the structure of self-dual codes, Type I codes and Type II codes over the ring <span class="math inline">\(R_{u^3}\)</span> in terms of the structure of their Torsion and Residue codes. We construct Type I and Type II optimal codes over the ring <span class="math inline">\(R_{u^3}\)</span> for different lengths.</p>