On the construction of self-complementary codes and their application in the problem of information hiding

Line codes are widely used to protect against errors in data transmission and storage systems, to ensure the stability of various cryptographic algorithms and protocols, to protect hidden information from errors in a stegocontainer. One of the classes of codes that find application in a number of the listed areas is the class of linear self-complementary codes over a binary field. Such codes contain a vector of all ones, and their weight enumerator is a symmetric polynomial. In applied problems, self-complementary $[n, k]$-codes are often required for a given length $n$ and dimension $k$ to have the maximum possible code distance $d(k, n)$. For $n < 13$, the values of $d(k, n)$ are already known. In this paper, for self-complementary codes of length $n=13, 14, 15$, the problem is to find lower bounds on $d(k, n)$, as well as to find the values of $d(k, n)$ themselves. The development of an efficient method for obtaining a lower estimate close to $d(k, n)$ is an urgent task, since finding the values of $d(k, n)$ in the general case is a difficult task. The paper proposes four methods for finding lower bounds: based on cyclic codes, based on residual codes, based on the $(u|u+v)$-construction, and based on the tensor product of codes. On the joint use of these methods for the considered lengths, it was possible to efficiently obtain lower bounds, either coinciding with the found values of $d(k, n)$ or differing by one. The paper proposes a sequence of checks, which in some cases helps to prove the absence of a self-complementary $[n, k]$-code with code distance $d$. In the final part of the work, on the basis of self-complementary codes, a design for hiding information is proposed that is resistant to interference in the stegocontainer. The above calculations show the greater efficiency of the new design compared to the known designs.