More on codes for combinatorial composite DNA

Abstract

In this paper, we focus on constructing unique-decodable and list-decodable codes for the recently studied (t, e)-composite-asymmetric error-correcting codes ((t, e)-CAECCs). Let \(\mathcal {X}\) be an \(m \times n\) binary matrix in which each row has Hamming weight w. If at most t rows of \(\mathcal {X}\) contain errors, and in each erroneous row, there are at most e occurrences of \(1 \rightarrow 0\) errors, we say that a (t, e)-composite-asymmetric error occurs in \(\mathcal {X}\). For general values of m, n, w, t, and e, we propose new constructions of (t, e)-CAECCs with redundancy at most \((t-1)\log (m) + O(1)\), where O(1) is independent of the code length m. In particular, this yields a class of (2, e)-CAECCs that are optimal in terms of redundancy. When m is a prime power, the redundancy can be further reduced to \((t-1)\log (m) - O(\log (m))\). To further increase the code size, we introduce a combinatorial object called a weak \(B_e\)-set. When \(e = w\), we present an efficient encoding and decoding method for our codes. Finally, we explore potential improvements by relaxing the requirement of unique decoding to list-decoding. We show that when the list size is t! or an exponential function of t, there exist list-decodable (t, e)-CAECCs with constant redundancy. When the list size is two, we construct list-decodable (3, 2)-CAECCs with redundancy \(\log (m) + O(1)\).