Construction of high rate $$\mathcal {T}$$ T - $${{\varvec{direct}}}$$ direct codes

Abstract

This paper presents a class of multi-user codes named as \(\mathcal {T}\)-direct tertiary codes. These classes of codes have higher information rate in comparison to the existing \(\mathcal {T}\)-direct codes. A coding scheme (encoding and decoding) is proposed for the class of \(\mathcal {T}\)-direct codes, that increases the information rate of the constituent codes in comparison to the existing coding scheme for the class of \(\mathcal {T}\)-direct codes: the coding scheme first drives the code \(\mathcal {C}_{i}'(n', k', d')\) from the constituent code \(\mathcal {C}_{i}(n, k, d)\) of existing \(\mathcal {T}\)-direct code \((\mathcal {C}_{1}, \mathcal {C}_{2}, \ldots , \mathcal {C}_{\mathcal {T}})\) with \(\frac{k'}{n'} \ge \frac{k}{n}\), and eventually forms a \(\mathcal {T}\)-direct code \((\mathcal {C}_{1}', \mathcal {C}_{2}', \ldots , \mathcal {C}_{\mathcal {T}}')\) with an overall increase in sum-rate. Just like \(\mathcal {T}\)-direct codes, the newly constructed class of codes can be used for simultaneous transmission of non-binary symbols over an \(\mathbb {F}\)-adder channel. Further, an associated decoding technique for the constructed class of codes is also proposed.